commatose

Hi, all,

A lot of recent discussions involving uses of commas left me realizing that
there is a whole bunch that I don't know. I've almost replied and asked
questions a dozen or so times recently, but ultimately I was too confused to
even come up with a question. So now I'm betting back to the latent realm
of questioning...

I took a scan through the tuning dictionary and I really didn't find what I
was hoping to see, which is kind of a summary of all the different ways that
commas can be used.

(Gee Monz, its not your fault -- its just that all my desires for knowledge
now get projected onto the tuning dictionary.)

Commas can be "left in" (not necessarily a piece of terminology), "tempered
out", "distributed" and perhaps many other possibilities. It would be nice
to see all these possibilities explained in one place. This is closely tied
in with unison vector and tempering, but actually *tempering* is not
necessarily necessary, and furthermore temperament sometime happens when it
is not intended, as per a recent conversation with Carl in which he informed
me that even by just playing an approximate 7-limit interval that occurs in
a 5-limit scale I have invoked "temperament". Maybe its too much to ask but
I'd love to see this all tied together in one comprehensive tutorial!

It seemed to me also that in the process of defining an actual tuning that
another concept of comma is perhaps introduced, but according to Carl this
isn't called a comma. When distributing a comma, which I think implies a
unison vector and also implies that the comma is rational (whereas the
historical use does not imply that, right?), the mechanics of distributing
it are likely (but not guaranteed I don't think) to introduce irrational
distributions among the various dimensions of the lattice. This
distributing reminds me of commas again, smaller commas, and my original
nieve intepretation of commas in temperament involved this perhaps
misfounded concept. I'm pretty vague on this, and welcome feedback.

The tonalsoft comma page currently relies heavily on a quote from Dave
Keenan which gives two definitions of comma. However neither definition
seems to say much about the functions of commas as commonly discussed on
this list, nor is any restriction to rational values mentioned. It seems to
me these comma functions apply regardless of the size of the comma, even if
they may seem a little absurd in some cases. So I'd almost wish there was
another word for this use of "comma", and it doesn't quite look like
"anomaly" is that word. But we could also go on with this usage and when
necessary invoke some term like "generalized comma" to indicate that a comma
has gotten outside of its normal size range, and perhaps "functional comma"
to indicate that it is the function and not the size that is at stake.
Around here everyone will usually know this from context. Nonetheless for
interfacing with a larger world we may need some kind of "qualification"
terms which can be invoked when necessary, probably rarely.

Has Paul dealt with this in his recent writings? I haven't had the time to
look at any of it yet.

Just doing my newbie job. ;)

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> I took a scan through the tuning dictionary and I really didn't
find what I
> was hoping to see, which is kind of a summary of all the different
ways that
> commas can be used.
...
> Commas can be "left in" (not necessarily a piece of
terminology), "tempered
> out", "distributed" and perhaps many other possibilities.

"Tempered out", "distributed", and "made to vanish" are all the same
thing in regard to commas (and schismas, kleismas, dieses etc. if we
disallow the generic use of "comma").

By the way "anomaly" doesn't work for me as a substitute for this
generic use of "comma" because of its implication of "a failure to
meet expectations".

If a comma (generic sense) is not tempered out (tempered to zero),
then tempering may make it larger, smaller or even negative,
relative to its untempered size, or it may remain untempered.

If it is "left in" then it may be an actual scale step (not likely,
but more likely for dieses and larger), or it may be a chromatic
alteration of a scale notes when modulation occurs, or it may be
symbolised as an accidental and used to notate scale notes even when
the note without that accidental never occurs. Or it may not
explicitly appear at all.

There is a borderline case where although there is no tempering, a
comma may already be so small that it can be ignored. So although it
vanishes it isn't exactly _made_ to vanish. This is most likely for
kleismas and schismas and smaller.

> It seemed to me also that in the process of defining an actual
tuning that
> another concept of comma is perhaps introduced, but according to
Carl this
> isn't called a comma. When distributing a comma, which I think
implies a
> unison vector and also implies that the comma is rational (whereas
the
> historical use does not imply that, right?),

Commas are always rational, ... until they are tempered. Hmm. That
doesn't sound too clear. :-) A comma is always _defined_ as a
rational frequency ratio, or equivalently as a prime-exponent-
vector. You can't just pick any random number of cents and call it a
comma.

> the mechanics of distributing
> it are likely (but not guaranteed I don't think) to introduce
irrational
> distributions among the various dimensions of the lattice.

Correct.

> This
> distributing reminds me of commas again, smaller commas, and my
original
> nieve intepretation of commas in temperament involved this perhaps
> misfounded concept. I'm pretty vague on this, and welcome
feedback.
>

You're right. It is misfounded. The quarter-comma of quarter-comma
meantone, is not itself a comma, because it doesn't arise as the
difference between two different stacks of rational intervals, and
cannot be defined in rational terms. The fourth root of 81/80 is
irrational.

> The tonalsoft comma page currently relies heavily on a quote from
Dave
> Keenan which gives two definitions of comma. However neither
definition
> seems to say much about the functions of commas as commonly
discussed on
> this list,

No, nothing about functions, sorry.

> nor is any restriction to rational values mentioned.

Technically, you're right! Yikes! But it was certainly my intention
to so restrict it.

Monz, if you could change those two ocurrences of "pitch ratios"
to "rational pitches" that should solve the problem.

> It seems to
> me these comma functions apply regardless of the size of the
comma, even if
> they may seem a little absurd in some cases.

Agreed. But I did include the deliberately vague
proscription "typically smaller than a scale step".

> So I'd almost wish there was
> another word for this use of "comma",

Me too. One suggestion I have is "komma".

But whether using "komma" or "comma" to cover this broader range of
sizes that have the same range of functions, it is well to spell out
the fact that you are doing so.

> and it doesn't quite look like
> "anomaly" is that word.

Agreed.

> But we could also go on with this usage and when
> necessary invoke some term like "generalized comma" to indicate
that a comma
> has gotten outside of its normal size range, and
perhaps "functional comma"
> to indicate that it is the function and not the size that is at
stake.

Those sound reasonable to me. The biggest disagreement is the one
about what functions a comma may serve and still be a comma (or
still be commatic). I haven't yet seen anything that would make me
accept that "chromatic comma" (non-vanishing comma) is some kind of
oxymoron.

> Around here everyone will usually know this from context.
Nonetheless for
> interfacing with a larger world we may need some kind
of "qualification"
> terms which can be invoked when necessary, probably rarely.
>
> Has Paul dealt with this in his recent writings? I haven't had
the time to
> look at any of it yet.

Paul Erlich has followed Paul Hahn in using "commatic" as the
opposite of "chromatic". The _sound_ of these two words is certainly
seductive in making one want them as opposites. "chromatic"
versus "vanishing" certainly doesn't sound as nice. And the other
seductive fact is that for about the last 400 years in the west, the
most important comma, the syntonic comma, has been something that
vanishes, and for about the last 100 years, the second most
important comma, the Pythagorean comma, has also been something that
vanishes.

But the fact is that both of these were originally named as commas
at a time when temperament was unknown.

> Just doing my newbie job. ;)

And a great job it is!

Hi Kurt,

>Maybe its too much to ask but
>I'd love to see this all tied together in one comprehensive tutorial!

Have you read Paul's papers?

>It seemed to me also that in the process of defining an actual tuning that
>another concept of comma is perhaps introduced, but according to Carl this
>isn't called a comma.

As you can see from the recent threads here, there is anything but
agreement on what a "comma" is. Nevertheless, the language seems to
work among those who already understand the concepts.

>When distributing a comma, which I think implies a
>unison vector

As it happens, there's also very little agreement on what a
"unison vector" is.

>and also implies that the comma is rational (whereas the
>historical use does not imply that, right?),

It may be said that commas are *usually* rational.

>The tonalsoft comma page currently relies heavily on a quote from Dave
>Keenan which gives two definitions of comma. However neither definition
>seems to say much about the functions of commas as commonly discussed on
>this list, nor is any restriction to rational values mentioned.

It's clear we've a long way to a consensus on this stuff, let alone
a clear tutorial on it. My advice is to ignore terminology, and
instead start working out tunings that interest you. In the process
you will aquire the tools you need.

-Carl

hi Kurt and Dave,

first i want to say: Kurt, that is one of the
greatest subject-lines i've ever seen around here. :)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > I took a scan through the tuning dictionary and I
> > really didn't find what I was hoping to see, which is
> > kind of a summary of all the different ways that
> > commas can be used.
> ...
> > Commas can be "left in" (not necessarily a piece of
> > terminology), "tempered out", "distributed" and perhaps
> > many other possibilities.
>
> "Tempered out", "distributed", and "made to vanish" are
> all the same thing in regard to commas (and schismas,
> kleismas, dieses etc. if we disallow the generic use of
> "comma").

i should say now that the "temper" page in the Encyclopaedia

http://tonalsoft.com/enc/index2.htm?temper.htm

which includes "temper out (vanish)" as its "part 2",
was very quickly thrown together, just so that i had
a page that defined the concept. there is a *lot*
(and i mean *really* a lot) more that i can add there.

i try to make a page for everything that i think
should be there, but even so, there are still some
very important terms that have not made it into
the Encyclopaedia. "flat" and "sharp" only just
got their pages a few weeks ago (5 years after the
inception of the Dictionary) -- and those two terms
are both pretty darn basic to tuning!

guess i was just too busy dealing with "periodicity-block"
and "unison-vector" and "val" and "proslambanomonos" ...

;-)

> By the way "anomaly" doesn't work for me as a substitute
> for this generic use of "comma" because of its implication
> of "a failure to meet expectations".

i never really liked "anomaly" either ... i got it
from Wilkinson, and he does indeed use it to describe
small JI intervals which fail to close the system,
which i suppose is an EDO-centric viewpoint and
therefore really shouldn't be extended to JI.

but i really don't like the use of "comma" in its
more general sense, precisely *because* it has such
a long history of being used to designate an interval
of a particular size-range.

"diesis" has been already suffered great variability
in meaning over the course of its history, and now
*we* are doing the same thing to "comma".

i would much like to have a general term to refer
to *all* of these small intervals -- schisminas,
skhismas, kleismas, commas, dieses, and whatever else.
but let's come up with a new one, not create a
secondary definition for "comma" (well, i guess that's
already a done deal ... but i for one will not advocate
its use with the general meaning).

if "anomaly" is unacceptable or unpopular, fine
-- so let's think up something else that's good.

> There is a borderline case where although there is no tempering,
> a comma may already be so small that it can be ignored. So
> although it vanishes it isn't exactly _made_ to vanish. This
> is most likely for kleismas and schismas and smaller.

and that is *precisely* what Fokker meant by "unison-vector".
so that is the term which should be used to designate that case.

> > It seemed to me also that in the process of defining
> > an actual tuning that another concept of comma is perhaps
> > introduced, but according to Carl this isn't called a comma.
> > When distributing a comma, which I think implies a
> > unison vector and also implies that the comma is rational
> > (whereas the historical use does not imply that, right?),
>
> Commas are always rational, ... until they are tempered.
> Hmm. That doesn't sound too clear. :-) A comma is always
> _defined_ as a rational frequency ratio, or equivalently
> as a prime-exponent-vector. You can't just pick any random
> number of cents and call it a comma.

Dave, your last sentence is true. but -- and i'm assuming
you're using "comma" in the general sense here -- you
certainly can find a "comma" which exists between pitches
at the two extremes of a meantone chain. those two pitches
have irrational "ratios", and so does that "comma".

for example, the interval between the origin and the
8ve-reduced 31st generator of 1/4-comma meantone, is
2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"
is definitely an irrational number ... and in fact,
is the interval which vanishes in 31edo, and also
which is *why* 31edo is such a good emulation of
1/4-comma meantone.

> > The tonalsoft comma page currently relies heavily on
> > a quote from Dave Keenan which gives two definitions of
> > comma. However neither definition seems to say much
> > about the functions of commas as commonly discussed on
> > this list,
>
> No, nothing about functions, sorry.
>
> > nor is any restriction to rational values mentioned.
>
> Technically, you're right! Yikes! But it was certainly my
> intention to so restrict it.
>
> Monz, if you could change those two ocurrences of
> "pitch ratios" to "rational pitches" that should solve
> the problem.

well ... i don't think that's a good idea, as per my
discussion above.

we need to discuss this more before i change that
definition. perhaps it's a good idea to have separate
terms to differentiate between rational "commas" and
irrational ones.

(there i go, encouraging the proliferation of jargon again ...)

> > It seems to me these comma functions apply regardless
> > of the size of the comma, even if they may seem a
> > little absurd in some cases.
>
> Agreed. But I did include the deliberately vague
> proscription "typically smaller than a scale step".
>
> > So I'd almost wish there was
> > another word for this use of "comma",
>
> Me too. One suggestion I have is "komma".
>
> But whether using "komma" or "comma" to cover this
> broader range of sizes that have the same range of
> functions, it is well to spell out the fact that you
> are doing so.

if you recall, i too at first accepted the komma/comma
difference in spelling to represent the difference
in meaning ... but gave it up in the face of protest
from others ... and i largely agreed with them,
because of the similarity of spelling.

a totally new word would be better, and i will
continue to maintain that position in the face
of all contrary arguments.

> > and it doesn't quite look like
> > "anomaly" is that word.
>
> Agreed.

as i said, i don't really care for "anomaly" either
... but it *does* get across the meaning of a small
rational interval which almost-but-not-quite closes
the tuning into a circle.

> > But we could also go on with this usage and when
> > necessary invoke some term like "generalized comma"
> > to indicate that a comma has gotten outside of its
> > normal size range, and perhaps "functional comma"
> > to indicate that it is the function and not the size
> > that is at stake.
>
> Those sound reasonable to me.

i still have to disagree. IMO, it's always better
to have a single short word to represent a concept
instead of a more general word with a preceding
qualifier which narrows its meaning.

i'll say it again: new terms, however strange they
seem at first, become familiar with usage!

every language that has enough speakers/writers to
keep it alive will always keep growing. new words
are invented all the time ... what really good
argument can anyone present against that? sorry,
i just won't accept the "too hard for newbies" line.

my Encyclopaedia is never further than a mouse-click
away, and if the word isn't in there or isn't well-defined,
then it should be and eventually will be.

> The biggest disagreement
> is the one about what functions a comma may serve and
> still be a comma (or still be commatic). I haven't yet
> seen anything that would make me accept that "chromatic comma"
> (non-vanishing comma) is some kind of oxymoron.

the biggest problem i've always had with "chromatic
unison-vector" or "chromatic comma" is that the Greek
root "chromatic" refers to scale elements which are
approximately a semitone apart, and i don't want to
see that "chromatic" referring to intervals which are
much smaller than that -- which *is* very possible
according to the recent usages i've seen.

and as i've said, the Greek root "chroma" already
has a significantly different meaning (essentially
the same as "pitch-class") ... so this could get messy,
but i will do everything i can to prevent that.

> > Around here everyone will usually know this from
> > context. Nonetheless for interfacing with a larger
> > world we may need some kind of "qualification"
> > terms which can be invoked when necessary, probably
> > rarely.
> >
> > Has Paul dealt with this in his recent writings?
> > I haven't had the time to look at any of it yet.
>
> Paul Erlich has followed Paul Hahn in using "commatic" as
> the opposite of "chromatic". The _sound_ of these two words
> is certainly seductive in making one want them as opposites.
> "chromatic" versus "vanishing" certainly doesn't sound as nice.
> And the other seductive fact is that for about the last
> 400 years in the west, the most important comma, the
> syntonic comma, has been something that vanishes, and for
> about the last 100 years, the second most important comma,
> the Pythagorean comma, has also been something that
> vanishes.
>
> But the fact is that both of these were originally named
> as commas at a time when temperament was unknown.

which strengthens my argument that we really need a
term to designate the "vanishingness" of a small interval.

i propose "vapro", for "VAnishing PROmo".

and i also like the fact that it brings to mind "vapor",
which is sort of like what those small vanishing intervals
become.

(and for those who don't like "promo" ... too bad.
in my recent discussions with Paul it has now become an
essential term, and i'm quite certain the rest of you
will eventually see why.)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Dave, your last sentence is true. but -- and i'm assuming
> you're using "comma" in the general sense here -- you
> certainly can find a "comma" which exists between pitches
> at the two extremes of a meantone chain. those two pitches
> have irrational "ratios", and so does that "comma".
>
> for example, the interval between the origin and the
> 8ve-reduced 31st generator of 1/4-comma meantone, is
> 2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"
> is definitely an irrational number ... and in fact,
> is the interval which vanishes in 31edo, and also
> which is *why* 31edo is such a good emulation of
> 1/4-comma meantone.

I would not suggest defining "promo" in a way which excludes this
example. If you take a monzo |A B C>, the promo is defined just by the
ratios A:B:C. You are talking about the same thing really whether A,
B, and C are restricted to being just integers, or are allowed to be
rational numbers. Traditionally, you let them be rational; it makes
the math a little easier. That would mean your "prime space" in which
a promo is said to be a "line" actually would be a (rational) vector
space, and the line actually would be a line. The 18:0:-31/4 ratios
and the 72:0:-31 ratios would then be simply two ways of denoting the
same promo.

> > Monz, if you could change those two ocurrences of
> > "pitch ratios" to "rational pitches" that should solve
> > the problem.

> well ... i don't think that's a good idea, as per my
> discussion above.

I suggest allowing them, at least, to denote promos.

> we need to discuss this more before i change that
> definition. perhaps it's a good idea to have separate
> terms to differentiate between rational "commas" and
> irrational ones.

Remember that we would be talking of a very special case of irrational
"commas" if we allowed fractional exponents. If we did the same
business with the Wilson fifth and 69-et, we would not end up with
something which could be discussed in terms of rational exponents, yet
2^40/Wilson^69, an interval of 1.5 cents, is certainly small enough to
be considered comatonse.

> > Me too. One suggestion I have is "komma".
> >
> > But whether using "komma" or "comma" to cover this
> > broader range of sizes that have the same range of
> > functions, it is well to spell out the fact that you
> > are doing so.

"Komma" is not a usable word in my book since it is simply an
alternative spelling, with the same sound.

> i propose "vapro", for "VAnishing PROmo".

I'd prefer just to say "promo", unless you forsee a use for projective
monzos which do not vanish.

on 8/13/04 10:56 PM, Dave Keenan <d.keenan@bigpond.net.au> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> I took a scan through the tuning dictionary and I really didn't
> find what I
>> was hoping to see, which is kind of a summary of all the different
> ways that
>> commas can be used.
> ...
>> Commas can be "left in" (not necessarily a piece of
> terminology), "tempered
>> out", "distributed" and perhaps many other possibilities.
>
> "Tempered out", "distributed", and "made to vanish" are all the same
> thing in regard to commas (and schismas, kleismas, dieses etc. if we
> disallow the generic use of "comma").

What do you mean by "same thing"? There are differences between these
things, right? Just checking. I think you mean something like as far as
the comma functions in defining the topology (?) of the resulting scale they
are all the same. But the topology (or whatever you want to call it) does
not determine the tuning. Right?

> By the way "anomaly" doesn't work for me as a substitute for this
> generic use of "comma" because of its implication of "a failure to
> meet expectations".
>
> If a comma (generic sense) is not tempered out (tempered to zero),
> then tempering may make it larger, smaller or even negative,
> relative to its untempered size, or it may remain untempered.

Making it larger seems an unlikely thing if you are going to collapse the
topology on that comma. The word tempering always implies a tuning, right?
Whereas a temperament as it is often used here is a class of tunings that
share a topology, right?

> If it is "left in" then it may be an actual scale step (not likely,
> but more likely for dieses and larger), or it may be a chromatic
> alteration of a scale notes when modulation occurs, or it may be
> symbolised as an accidental and used to notate scale notes even when
> the note without that accidental never occurs. Or it may not
> explicitly appear at all.
>
> There is a borderline case where although there is no tempering, a
> comma may already be so small that it can be ignored. So although it
> vanishes it isn't exactly _made_ to vanish. This is most likely for
> kleismas and schismas and smaller.

Yet if a piece makes use of this then the piece makes the comma vanish,
structurally speaking?

>> It seemed to me also that in the process of defining an actual
> tuning that
>> another concept of comma is perhaps introduced, but according to
> Carl this
>> isn't called a comma. When distributing a comma, which I think
> implies a
>> unison vector and also implies that the comma is rational (whereas
> the
>> historical use does not imply that, right?),
>
> Commas are always rational, ... until they are tempered. Hmm. That
> doesn't sound too clear. :-) A comma is always _defined_ as a
> rational frequency ratio, or equivalently as a prime-exponent-
> vector. You can't just pick any random number of cents and call it a
> comma.

Ok, I'll be interested to see how you reply to Monz's issue about this.

>> This
>> distributing reminds me of commas again, smaller commas, and my
> original
>> nieve intepretation of commas in temperament involved this perhaps
>> misfounded concept. I'm pretty vague on this, and welcome
> feedback.
>>
>
> You're right. It is misfounded. The quarter-comma of quarter-comma
> meantone, is not itself a comma, because it doesn't arise as the
> difference between two different stacks of rational intervals, and
> cannot be defined in rational terms. The fourth root of 81/80 is
> irrational.

Ah, is *that* what distributing usually refers to? I was thinking of
distributing the tempering among multiple dimensions of a lattice, but this
is distributing in a single dimension (though in anotheer sense multiple
dimensions are involved).

>> It seems to
>> me these comma functions apply regardless of the size of the
> comma, even if
>> they may seem a little absurd in some cases.
>
> Agreed. But I did include the deliberately vague
> proscription "typically smaller than a scale step".
>
>> So I'd almost wish there was
>> another word for this use of "comma",
>
> Me too. One suggestion I have is "komma".

If this functioning is primarily topological, isn't there already a
mathematical term for collapsing a topology according to a certain pattern,
and for the specification of the pattern over which the collapse is done?
Gene?

>> But we could also go on with this usage and when
>> necessary invoke some term like "generalized comma" to indicate
> that a comma
>> has gotten outside of its normal size range, and
> perhaps "functional comma"
>> to indicate that it is the function and not the size that is at
> stake.
>
> Those sound reasonable to me. The biggest disagreement is the one
> about what functions a comma may serve and still be a comma (or
> still be commatic). I haven't yet seen anything that would make me
> accept that "chromatic comma" (non-vanishing comma) is some kind of
> oxymoron.
>
>> Around here everyone will usually know this from context.
> Nonetheless for
>> interfacing with a larger world we may need some kind
> of "qualification"
>> terms which can be invoked when necessary, probably rarely.
>>
>> Has Paul dealt with this in his recent writings? I haven't had
> the time to
>> look at any of it yet.
>
> Paul Erlich has followed Paul Hahn in using "commatic" as the
> opposite of "chromatic". The _sound_ of these two words is certainly
> seductive in making one want them as opposites. "chromatic"
> versus "vanishing" certainly doesn't sound as nice.

Isn't "chromatic" also asking for trouble by still restricting the size?
Didn't Paul or someone come up with an example a couple months back of a
whole-tone sized comma?

Thanks for your inputs.

-Kurt

on 8/13/04 11:11 PM, Carl Lumma <ekin@lumma.org> wrote:

> Hi Kurt,
>
>> Maybe its too much to ask but
>> I'd love to see this all tied together in one comprehensive tutorial!
>
> Have you read Paul's papers?

I haven't had the time yet. Should I be quiet until I do? (I may be quite
for a long time. ;)

>> When distributing a comma, which I think implies a
>> unison vector
>
> As it happens, there's also very little agreement on what a
> "unison vector" is.

Yes, I think I need to start another thread about the vector issue.

> It's clear we've a long way to a consensus on this stuff, let alone
> a clear tutorial on it. My advice is to ignore terminology, and
> instead start working out tunings that interest you. In the process
> you will aquire the tools you need.

Yes, I think you're right. Like many others when first exposed to lingo
that is not well worked-out, I get confused trying to learn, and then I have
a little reaction to the chaos. But in the end I'm fine with ill-defined
terms because in the end all our words can appear insufficient for our
experiences, and so the art is in overcoming those apparent limits.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/13/04 11:11 PM, Carl Lumma <ekin@l...> wrote:
> > It's clear we've a long way to a consensus on this stuff, let
alone
> > a clear tutorial on it. My advice is to ignore terminology, and
> > instead start working out tunings that interest you. In the
process
> > you will aquire the tools you need.
>
> Yes, I think you're right. Like many others when first exposed to
lingo
> that is not well worked-out, I get confused trying to learn, and
then I have
> a little reaction to the chaos. But in the end I'm fine with ill-
defined
> terms because in the end all our words can appear insufficient for
our
> experiences, and so the art is in overcoming those apparent limits.

Carl and Kurt,

These are wise words indeed. Thank you both.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > Commas are always rational, ... until they are tempered.
> > Hmm. That doesn't sound too clear. :-) A comma is always
> > _defined_ as a rational frequency ratio, or equivalently
> > as a prime-exponent-vector. You can't just pick any random
> > number of cents and call it a comma.
>
>
>
> Dave, your last sentence is true. but -- and i'm assuming
> you're using "comma" in the general sense here -- you
> certainly can find a "comma" which exists between pitches
> at the two extremes of a meantone chain. those two pitches
> have irrational "ratios", and so does that "comma".
>
> for example, the interval between the origin and the
> 8ve-reduced 31st generator of 1/4-comma meantone, is
> 2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"
> is definitely an irrational number ... and in fact,
> is the interval which vanishes in 31edo, and also
> which is *why* 31edo is such a good emulation of
> 1/4-comma meantone.
>
>
>
> > > The tonalsoft comma page currently relies heavily on
> > > a quote from Dave Keenan which gives two definitions of
> > > comma. However neither definition seems to say much
> > > about the functions of commas as commonly discussed on
> > > this list,
> >
> > No, nothing about functions, sorry.
> >
> > > nor is any restriction to rational values mentioned.
> >
> > Technically, you're right! Yikes! But it was certainly my
> > intention to so restrict it.
> >
> > Monz, if you could change those two ocurrences of
> > "pitch ratios" to "rational pitches" that should solve
> > the problem.
>
>
> well ... i don't think that's a good idea, as per my
> discussion above.

But the comma you refer to is tempered. That's the only reason it's
irrational. Untempered, it may be expressed as 3^31 / 2^49 or any
number of other rationals. So this is not a counterexample.

> a totally new word would be better, and i will
> continue to maintain that position in the face
> of all contrary arguments.

That sounds sadly like dogmatism or fanaticism.

> which strengthens my argument that we really need a
> term to designate the "vanishingness" of a small interval.
>
> i propose "vapro", for "VAnishing PROmo".

Ah Monz, ... you're a lunatic ... but a lovable one. :-)

Newbie: What's a "vapro"?

Old hand: It's a "vanishing promo"?

Newbie: What's a "promo"?

Old hand: Well it's either something you do to promote yourself, or
it's a "projective monzo".

Newbie: What's a "monzo"?

Old hand: Well it's either a prime exponent vector, or it's one of
the guys that made up this jargon.

Newbie: Oh. Now I get why it's a promo.

;-)

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> > "Tempered out", "distributed", and "made to vanish" are all the
same
> > thing in regard to commas (and schismas, kleismas, dieses etc.
if we
> > disallow the generic use of "comma").
>
> What do you mean by "same thing"? There are differences between
these
> things, right?

Not that I can think of. Commas never really vanish, they either get
spread so thinly (by being distributed over multiple intervals) so
that you don't notice them, or you _decide_ not to notice them.
That's what tempering is.

> Just checking. I think you mean something like as far as
> the comma functions in defining the topology (?) of the resulting
scale they
> are all the same. But the topology (or whatever you want to call
it) does
> not determine the tuning. Right?

You've lost me here. Sorry.

> Making it larger seems an unlikely thing if you are going to
collapse the
> topology on that comma.

Sure. But who said we were? When we make some commas vanish, others
will get larger.

> The word tempering always implies a tuning, right?

Not sure I understand the question? Tempering is the process of
ditributing commas and the result is a temperament.

> Whereas a temperament as it is often used here is a class of
tunings that
> share a topology, right?

I guess so.

> Yet if a piece makes use of this then the piece makes the comma
vanish,
> structurally speaking?

I guess so. Like I said, it's borderline.

> Ah, is *that* what distributing usually refers to? I was thinking
of
> distributing the tempering among multiple dimensions of a lattice,
but this
> is distributing in a single dimension (though in anotheer sense
multiple
> dimensions are involved).

You're quite right in both cases. This might help:

http://dkeenan.com/Music/DistributingCommas.htm

> Isn't "chromatic" also asking for trouble by still restricting the
size?

I don't see it as restricting the size. But you may be right.

We're generalising from "chromatic" meaning "not in the diatonic
scale" or "indicated by a sharp or flat accidental outside of the
key signature" to anything that is not in the "usual" scale of a
given temperament and is indicated by _any_ accidental, not just a
sharp or flat.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> > > > The tonalsoft comma page currently relies heavily on
> > > > a quote from Dave Keenan which gives two definitions of
> > > > comma. However neither definition seems to say much
> > > > about the functions of commas as commonly discussed on
> > > > this list,
> > >
> > > No, nothing about functions, sorry.
> > >
> > > > nor is any restriction to rational values mentioned.
> > >
> > > Technically, you're right! Yikes! But it was certainly my
> > > intention to so restrict it.
> > >
> > > Monz, if you could change those two ocurrences of
> > > "pitch ratios" to "rational pitches" that should solve
> > > the problem.
> >
> >
> > well ... i don't think that's a good idea, as per my
> > discussion above.
>
> But the comma you refer to is tempered. That's the only reason it's
> irrational. Untempered, it may be expressed as 3^31 / 2^49 or any
> number of other rationals. So this is not a counterexample.

hmm ... ok, well ... in any case, it's far to late (early?)
for me to think about this stuff anymore "tonight".
my brain is fried.

i think we need some more discussion on this one.

> > a totally new word would be better, and i will
> > continue to maintain that position in the face
> > of all contrary arguments.
>
> That sounds sadly like dogmatism or fanaticism.

yeah, upon reading it again it sounds like that
to me too. thanks for calling me on that.

i didn't really have to make such a strong stand
anyway ... all i had to do was make up a word i like
and put it into the Encyclopaedia. ;-PP

... which is exactly what i've done anyway.

> > which strengthens my argument that we really need a
> > term to designate the "vanishingness" of a small interval.
> >
> > i propose "vapro", for "VAnishing PROmo".
>
> Ah Monz, ... you're a lunatic ... but a lovable one. :-)

kisses all around from me too!

> Newbie: What's a "vapro"?
>
> Old hand: It's a "vanishing promo"?
>
> Newbie: What's a "promo"?
>
> Old hand: Well it's either something you do to promote yourself, or
> it's a "projective monzo".
>
> Newbie: What's a "monzo"?
>
> Old hand: Well it's either a prime exponent vector, or it's one of
> the guys that made up this jargon.
>
> Newbie: Oh. Now I get why it's a promo.
>
> ;-)

ok, wiseguy ... how about:

Newbie: What's a "vapro"?

Old hand:
http://tonalsoft.com/enc/index2.htm?vapro.htm

Newbie: What's a "promo"?

Old hand:
http://tonalsoft.com/enc/index2.htm?promo.htm

Newbie: What's a "monzo"?

Old hand:
http://tonalsoft.com/enc/index2.htm?monzo.htm

if i had somebody around who could have given me
answers like that 10 years ago, i would have saved
myself a ridiculous amount of time and effort.

-monz

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > > "Tempered out", "distributed", and "made to vanish"
> > > are all the same thing in regard to commas (and schismas,
> > > kleismas, dieses etc. if we disallow the generic use of
> > > "comma").
> >
> > What do you mean by "same thing"? There are differences
> > between these things, right?
>
> Not that I can think of. Commas never really vanish, they
> either get spread so thinly (by being distributed over
> multiple intervals) so that you don't notice them, or you
> _decide_ not to notice them. That's what tempering is.

i can't bring myself to say that they don't really vanish.

if one makes a lattice in a certain dimensionality of
prime-space, then distributes the promo ("comma") so
that it cannot be noticed (thus making it a vapro),
then warps the lattice to show the fact that it vanishes,
it really does physically vanish from the lattice!

that's exactly how Musica made these 4-strand helical
meantone lattices:

http://tonalsoft.com/enc/index2.htm?meantone.htm&helix

(wait for it to load all the way and it will take you
right to those lattices.)

> > Whereas a temperament as it is often used here is a
> > class of tunings that share a topology, right?
>
> I guess so.

i think that might be a good description of a temperament
"family". but i'll defer here to others who know more
about topology.

-monz

>> "Tempered out", "distributed", and "made to vanish" are all the same
>> thing in regard to commas (and schismas, kleismas, dieses etc. if we
>> disallow the generic use of "comma").
>
>What do you mean by "same thing"? There are differences between these
>things, right?

No. These phrases are usually used interchangeably.

>> There is a borderline case where although there is no tempering, a
>> comma may already be so small that it can be ignored. So although it
>> vanishes it isn't exactly _made_ to vanish. This is most likely for
>> kleismas and schismas and smaller.
>
>Yet if a piece makes use of this then the piece makes the comma vanish,
>structurally speaking?

Yes.

>> Commas are always rational, ... until they are tempered. Hmm. That
>> doesn't sound too clear. :-) A comma is always _defined_ as a
>> rational frequency ratio, or equivalently as a prime-exponent-
>> vector. You can't just pick any random number of cents and call it a
>> comma.

That's what Dave's saying. I'm not sure it's strictly true in
the literature, though.

>> You're right. It is misfounded. The quarter-comma of quarter-comma
>> meantone, is not itself a comma, because it doesn't arise as the
>> difference between two different stacks of rational intervals, and
>> cannot be defined in rational terms. The fourth root of 81/80 is
>> irrational.
>
>Ah, is *that* what distributing usually refers to? I was thinking of
>distributing the tempering among multiple dimensions of a lattice, but
>this is distributing in a single dimension (though in anotheer sense
>multiple dimensions are involved).

Distributing is distributing, any way you'd like to do it!

>>> But we could also go on with this usage and when necessary
>>> invoke some term like "generalized comma" to indicate
>>> that a comma has gotten outside of its normal size range,
>>> and perhaps "functional comma" to indicate that it is the
>>> function and not the size that is at stake.
>>
>> Those sound reasonable to me. The biggest disagreement is the one
>> about what functions a comma may serve and still be a comma (or
>> still be commatic). I haven't yet seen anything that would make me
>> accept that "chromatic comma" (non-vanishing comma) is some kind of
>> oxymoron.

"Chromatic comma" is fine. "Comma" does not usually imply vanishing,
though Gene wanted to make it so.

>Isn't "chromatic" also asking for trouble by still restricting the
>size? Didn't Paul or someone come up with an example a couple months
>back of a whole-tone sized comma?

Dave is one of a vast minority of theorists who worries about
defining rigid size constraints for things like commas.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> If this functioning is primarily topological, isn't there already a
> mathematical term for collapsing a topology according to a certain
pattern,
> and for the specification of the pattern over which the collapse is
done?
> Gene?

Don't ask me. Words such as "topology" and "topologicial" have
specific mathematical meanings, and the way you are using them is
making me dizzy. Can you explain what you mean more precisely, and
without using "topology" or "topological"?

> Isn't "chromatic" also asking for trouble by still restricting the size?
> Didn't Paul or someone come up with an example a couple months back of a
> whole-tone sized comma?

I've just employed 125/108 as a comma for a Fokker block with great
popular sucess; 125/108 is a half-fourth sized sub-subminor third of
253 cents.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> i can't bring myself to say that they don't really vanish.

But in any case, you agree that making them vanish, distributing
them, and tempering them out are all the same thing?

> if one makes a lattice in a certain dimensionality of
> prime-space, then distributes the promo ("comma") so
> that it cannot be noticed (thus making it a vapro),
> then warps the lattice to show the fact that it vanishes,
> it really does physically vanish from the lattice!

Yes. It vanishes from the lattice, but the lattice isn't physical,
it's a mathematical abstraction. What's physical is what you hear,
or measure in Hertz or cents.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> "Chromatic comma" is fine. "Comma" does not usually imply
vanishing,
> though Gene wanted to make it so.

Actually I think Gene got that from Paul Erlich, who got it from
Paul Hahn via the term "commatic" being juxtaposed
against "chromatic".

Also, Gene says his "kernel elements of a temperament" got
translated to "commas of a temperament", and in that context,
there's not too much potential for confusion since a temperament's
vanishing commas are more special than its chromatic ones. The
former can define the temperament whereas the latter are somewhat
arbitrary, depending as they do on a choice of a "white-note" scale.

> >Isn't "chromatic" also asking for trouble by still restricting the
> >size? Didn't Paul or someone come up with an example a couple
months
> >back of a whole-tone sized comma?
>
> Dave is one of a vast minority of theorists who worries about
> defining rigid size constraints for things like commas.

Yes. But that isn't relevant here. Kurt seems to be assuming
that "chromatic comma" means a comma around a semitone in size (so
generic comma, not the approx 20 cents range), but the intended
meaning of chromatic here is "functioning as an alteration in pitch
from a "standard" scale of some temperament, or even from
Pythagorean".

>> >Isn't "chromatic" also asking for trouble by still restricting the
>> >size? Didn't Paul or someone come up with an example a couple
>> >months back of a whole-tone sized comma?
>>
>> Dave is one of a vast minority of theorists who worries about
>> defining rigid size constraints for things like commas.
>
>Yes. But that isn't relevant here. Kurt seems to be assuming
>that "chromatic comma" means a comma around a semitone in size (so
>generic comma, not the approx 20 cents range), but the intended
>meaning of chromatic here is "functioning as an alteration in pitch
>from a "standard" scale of some temperament, or even from
>Pythagorean".

Yes, sorry about that. I misconstrued things.

-Carl

Gene,

on 8/14/04 11:47 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> If this functioning is primarily topological, isn't there already a
>> mathematical term for collapsing a topology according to a certain
> pattern,
>> and for the specification of the pattern over which the collapse is
> done?
>> Gene?
>
> Don't ask me. Words such as "topology" and "topologicial" have
> specific mathematical meanings, and the way you are using them is
> making me dizzy. Can you explain what you mean more precisely, and
> without using "topology" or "topological"?

Sure I can avoid this easily. I think we can approach what I was getting at
by defining "temperament" more clearly. The tonalsoft dictionary doesn't
seem to define it the way I expect it to be defined for usage in this group,
as I understand it, partly from talking to Carl about it recently.

Maybe I can start with a question. What uniquely determines a temperament?
Once I have this answer then I can continue with more clarity.

Well I can give more hints than that. This is about the distinction between
temperament and tuning. Traditionally a temperament was a tuning. On this
list, in relation to "comma theory" this is not the case. I think a
temperament is uniquely defined by the commas tempered out. But this does
not uniquely define a tuning, right? Once this is confirmed, I can go on.

-Kurt

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> if i had somebody around who could have given me
> answers like that 10 years ago, i would have saved
> myself a ridiculous amount of time and effort.

Yes Monz,

As I've said many times, I don't know what we'd do without your
encyclopedia. But don't you think that when an encyclopedia that's
edited by one person starts having entries for new terms added at
the whim of that editor and deleted again within a few days and new
terms put in their place, it seriously undermines the authority of
that encyclopedia. Why should anyone take any notice of any of it if
that sort of thing can happen?

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Maybe I can start with a question. What uniquely determines a
temperament?

I would say a regular temperament is uniquely determined by a
homomorphic mapping from the p-limit, or possibly another
finitely-generated subgroup of the positive rationals, to an abstract
free group of smaller rank. This can be specified by giving a wedgie,
a kernel, or an explicit mapping.

Note I do not include the tuning map as part of the definition, so
this is an abstract definition of what a regular temperament is.
However, this is how the word is most commonly used; people may object
to it but the same people talk of 1/4-comma meantone or 2/7-comma
meantone as if they were both meantone. My definition also says that
even though 31-et meantone is tuned to a group of rank one, it is
still qua meantone a group of rank two, and the tuning mapping is
another issue.

You might also note that this definition, which says a temperament is
a morphism, makes no sense unless you get the category right, which
connects to the thread about spaces.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> I would say a regular temperament is uniquely determined by a
> homomorphic mapping from the p-limit, or possibly another
> finitely-generated subgroup of the positive rationals, to an abstract
> free group of smaller rank. This can be specified by giving a wedgie,
> a kernel, or an explicit mapping.

I should have said it is determined by the mapping, and uniquely
determined by the wedgie or kernel.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > i can't bring myself to say that they don't really vanish.
>
> But in any case, you agree that making them vanish, distributing
> them, and tempering them out are all the same thing?
>
> > if one makes a lattice in a certain dimensionality of
> > prime-space, then distributes the promo ("comma") so
> > that it cannot be noticed (thus making it a vapro),
> > then warps the lattice to show the fact that it vanishes,
> > it really does physically vanish from the lattice!
>
> Yes. It vanishes from the lattice, but the lattice isn't physical,
> it's a mathematical abstraction. What's physical is what you hear,
> or measure in Hertz or cents.

and that is exactly *my* point. apparently we're
agreeing with each other from opposite sides of a tall
fence, which is blocking the view into each other's reasoning.

:)

for example,

you can listen to music in 12edo all day long and you
will never *hear* a syntonic-comma, or a diesis, or
a skhisma, or a pythagorean-comma.

yes, i composer might be able to figure out how to
play around with his listener's expectations so that
the listeners *think* they are going to hear one of
those small intervals ... but if the tuning is 12edo,
those intervals will never be heard because they
do not physically exist in that tuning.

in 5-limit JI, with an 8ve-equivalent lattice, you can
play the ratio which plots to 5^1 on the lattice
(ratio 5:4, monzo [-2 0, 1>) and you can play the
ratio which plots to 3^4 (ratio 81:64, monzo [-6 4, 0>),
and if you're listening carefully you will hear the
difference of a syntonic comma.

but in 12edo, both of these JI "targets" are mapped
to the same 12edo note. so no matter whether you
consider 2^(4/12) to map 3^4 or to map 5^1, listen
as carefully as you wish and you will never hear any
difference. the syntonic comma is not there,
and won't be found between any two pitches in 12edo.

-monz

>I would say a regular temperament is uniquely determined by a
>homomorphic mapping from the p-limit, or possibly another
>finitely-generated subgroup of the positive rationals, to an abstract
>free group of smaller rank. This can be specified by giving a wedgie,
>a kernel, or an explicit mapping.
//
>You might also note that this definition, which says a temperament is
>a morphism, makes no sense unless you get the category right, which
>connects to the thread about spaces.

Yes, you may indeed have a point here.

-Carl

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > Yes. It vanishes from the lattice, but the lattice isn't
physical,
> > it's a mathematical abstraction. What's physical is what you
hear,
> > or measure in Hertz or cents.
>
>
> and that is exactly *my* point. apparently we're
> agreeing with each other from opposite sides of a tall
> fence, which is blocking the view into each other's reasoning.
>
> :)
>
>
> for example,
>
> you can listen to music in 12edo all day long and you
> will never *hear* a syntonic-comma, or a diesis, or
> a skhisma, or a pythagorean-comma.

You can't hear a whole syntonic comma, because it's been cut into
pieces and spread around the place. But you can certainly hear these
pieces, not as melodic steps of course, but as roughness or beating
in the harmonies. So they have not actually _vanished_ from what you
hear.

...
> difference. the syntonic comma is not there,
> and won't be found between any two pitches in 12edo.

This is hardly the point. One can have a completely untempered JI
scale, e.g. Pythagorean 7 or 12, in which the syntonic comma can't
be found between any two pitches. We don't say that the syntonic
comma has been tempered out or made to vanish in this scale.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > Yes. It vanishes from the lattice, but the lattice
> > > isn't physical, it's a mathematical abstraction. What's
> > > physical is what you hear, or measure in Hertz or cents.
> >
> >
> > and that is exactly *my* point. apparently we're
> > agreeing with each other from opposite sides of a tall
> > fence, which is blocking the view into each other's reasoning.
> >
> > :)
> >
> >
> > for example,
> >
> > you can listen to music in 12edo all day long and you
> > will never *hear* a syntonic-comma, or a diesis, or
> > a skhisma, or a pythagorean-comma.
>
> You can't hear a whole syntonic comma, because it's been cut into
> pieces and spread around the place. But you can certainly hear these
> pieces, not as melodic steps of course, but as roughness or beating
> in the harmonies. So they have not actually _vanished_ from what you
> hear.
>
> ...
> > difference. the syntonic comma is not there,
> > and won't be found between any two pitches in 12edo.
>
> This is hardly the point. One can have a completely untempered JI
> scale, e.g. Pythagorean 7 or 12, in which the syntonic comma can't
> be found between any two pitches. We don't say that the syntonic
> comma has been tempered out or made to vanish in this scale.

but the syntonic-comma doesn't have anything to do
with a pythagorean tuning! you'll never find it
there either.

anyway, the thing you wrote about roughness in 12edo makes
your point clear. now i'm understanding you.

-monz

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > Maybe I can start with a question. What uniquely determines a
> temperament?

This is an excellent question.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I would say a regular temperament is uniquely determined by a
> homomorphic mapping from the p-limit, or possibly another
> finitely-generated subgroup of the positive rationals, to an
abstract
> free group of smaller rank. This can be specified by giving a
wedgie,
> a kernel, or an explicit mapping.

To many people on this list, like me, who haven't recently used or
studied abstract algebra, category theory, geometric algebra, and
Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as
meaningful as this:

"A regular temperament is uniquely determined by a
heteromagic mapping from the p-limit, or possibly another
finitely-gargled sub-band of the positive rationals, to an abstract
free band of lesser authority. This can be specified by pulling
someone's underpants real tight, a peanut, or an explicit mapping."

What if we started with the idea that a temperament was something
physical, that you can hear, or at least measure as a set of
frequencies.

The Shorter Oxford has this entry for the musical meaning
of "temperament":

"The adjustment of intervals of the scale (in the tuning of
instruments of fixed intonation, as keyboard instruments), so as to
adapt them to purposes of practical harmony : consisting in slight
variations of the pitch of the notes from true or 'just' intonation,
in order to make them available in different keys; a particular
system for doing this. (Sometimes extended to any system of tuning.)
1727."

The example text they quote is:

"The chief temperaments ... are mean-tone temperament and equal
temperament (now almost universal), in which the octave is divided
into twelve (theoretically) equal semitones, so that the variations
of pitch are evenly distributed throughout all keys."

We have generalised this idea beyond 5-limit and beyond 7-tone or 12-
tone chain-of-fifth scales in a fairly obvious way, about which
there is no dispute that I am aware of.

But something we haven't agreed on is the following recasting of
Kurt's question: "How can we tell, by listening or measuring,
whether two tunings are different temperaments, or are merely
slightly different tunings of the same temperament?"

To the mathematicians, the mapping from generators to prime number
intervals is everything. But you can't always obtain a unique
mapping by reverse-engineering the tuning. This was a very real
problem with claims made about the Zeng Bells a while back. More
than one mapping can result in the same tuning. Do we really want to
call them different temperaments if you can't hear or measure any
difference in the product? Maybe we do.

But notice that we don't do this with one dimensional (equal)
temperaments. Even something as simple as 24-ET has more than one
meaningful mapping for the primes number 7. But we don't refer to
these two mappings as two different temperaments. There is only one
24 tone equal temperament.

I will understand if you want to apply to me Bishop Berkeley's
observation on philosphers, "They first raise a dust, and then
complain that they cannot see." :-)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> What if we started with the idea that a temperament was something
> physical, that you can hear, or at least measure as a set of
> frequencies.

And when the Shorter Oxford Dictionary speaks of "meantone", what do
they mean? How some given piano is tuned on a given day--the Standard
Meantone Piano, held in a climate controlled room in Paris? When it
says the interval differences of equal temperament are "theoretically"
equal, is it not denying that some specific physical piano is being
talked about at all? "Theoretically equal" is an idealization. It
isn't something physical that you can hear, or measure as a set of
frequencies, it is a theoretical construct.

Using your system, you cannot name or classify temperaments. You can
have no theory, because you can have no theoretical constructs. You
can measure what a violinist does when he or she plays a score, but
the score itself is too abstract for your point of view to deal with;
the violinists who first played it may well be long dead, and no
recording of 18th century violin playing will allow you to measure
what was done in the 18th century.

> But notice that we don't do this with one dimensional (equal)
> temperaments. Even something as simple as 24-ET has more than one
> meaningful mapping for the primes number 7. But we don't refer to
> these two mappings as two different temperaments.

Wake up and smell the coffee. I certainly do, and I don't think I am
the only one.

There is only one
> 24 tone equal temperament.

There is *no* "24 tone equal temperament" by your logic. There are
only physically existing systems, producing sound by one means or
another, which may reasonably be considered instances of more or less
equal division of the octave into 24 parts. It is simply a way of
evaluating tunings. You can't ask what temperaments 24-equal supports,
because the question is meaningless. You can't write a score in it, as
to do that assumes quarter-tones are not necessarily physically
related to some particular place, time, and object. You certainly will
have grave difficulty coming up with any theory of temperaments;
including the sorts of theories you've put forward. Those are straight
out the door and in the trash now.

>> I would say a regular temperament is uniquely determined by a
>> homomorphic mapping from the p-limit, or possibly another
>> finitely-generated subgroup of the positive rationals, to an
>> abstract free group of smaller rank. This can be specified by
>> giving a wedgie, a kernel, or an explicit mapping.
>
>To many people on this list, like me, who haven't recently used or
>studied abstract algebra, category theory, geometric algebra, and
>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as
>meaningful as this:
>
>"A regular temperament is uniquely determined by a
>heteromagic mapping from the p-limit, or possibly another
>finitely-gargled sub-band of the positive rationals, to an abstract
>free band of lesser authority. This can be specified by pulling
>someone's underpants real tight, a peanut, or an explicit mapping."

ROTFL! :)

Though it meant a little more than this to me. Let me step
through Gene's definition...

regular temperament - The thing we're defining, which is
the target of a...

homomorphic mapping - I'd looked this up before, but I had to
do it again. It looks like this phrase could be replaced with
"homomorphism", since a morphism is already a type of mapping.
And homomorphism is just a "general morphism". It isn't clear
what kinds of mappings qualify as general. I'm assuming the
target of the map is expected to be the same sort of thing you
started with. For example, set -> set, group -> group, etc.
But it looks like we're dealing with groups here, so I think
Gene may have meant "group homomorphism", which means not only
do you wind up with a group, but that group has the same identity
element and the same operator as the group you started with.

...from the...

p-limit - Prime limit. Most list members are familiar with
these.

...or...

finitely-generated subgroup of the positive rationals - I
assume this allows for odd-limit and/or non-consective prime
bases like {2, 3, 7 11}.

...to an...

abstract free group of smaller rank - A group is a set that's
closed under some operator, has an identity element, and a law
of inverses. This was actually discussed in my 8th-grade
Algebra class. I'm not sure about the "free" part. It sounds
like it's meant to rule out torsion. The "abstract" part
looks superfluous to me. I think rank is just what we might
call dimension -- the number of generators needed, or more
precisely, the minimum number of elements needed to cover the
group using our operator. So by saying "smaller rank" we just
mean we expect temperament to simplify things.

...and can be named by any of...

wedgie - Gene and Graham have explained these. IIRC they
suck numbers out of the map and put them into a single
n-tuple in a certain way.

kernel - I haven't a clue.

explicit mapping - I think most tuning-math denizens are
familiar with these, though I've only worked with those
beloning to linear temperaments.

>But notice that we don't do this with one dimensional (equal)
>temperaments. Even something as simple as 24-ET has more than one
>meaningful mapping for the primes number 7. But we don't refer to
>these two mappings as two different temperaments. There is only one
>24 tone equal temperament.

Gene blew apart my thinking on this years ago, with this very
example, when I asked him why he doesn't care about consistency.
The answer is that all regular temperaments are consistent, and
that ETs are not necessarily temperaments. I haven't been able
to enjoy domestic beer since. :)

-Carl

Dave Keenan wrote:

> To the mathematicians, the mapping from generators to prime number > intervals is everything. But you can't always obtain a unique > mapping by reverse-engineering the tuning. This was a very real > problem with claims made about the Zeng Bells a while back. More > than one mapping can result in the same tuning. Do we really want to > call them different temperaments if you can't hear or measure any > difference in the product? Maybe we do.
> > But notice that we don't do this with one dimensional (equal) > temperaments. Even something as simple as 24-ET has more than one > meaningful mapping for the primes number 7. But we don't refer to > these two mappings as two different temperaments. There is only one > 24 tone equal temperament.

This is an interesting point: the difference between "bug" and "superpelog" for instance is mainly in the mapping, and in the number of generators in a typical scale. "Superpelog" is basically a 14-note scale with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2 3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is typical. But the generator / period ratio is very similar: 260.76 / 1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and 254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar tunings; the difference is in which notes are used to approximate consonant intervals. This is similar to the use of two different approximate fifths in ET's like 23-ET or 64-ET.

Another way to categorize tunings is by putting them on a branch of the scale tree (by their generator / period ratio if the period is around an octave). This would lump together scales that have different mappings, but sound roughly the same. For each specific generator / period ratio, a number of different mappings can be assigned depending on how far you continue the chain of generators in one direction or the other. One thing this could be useful for is notation: it would make sense to use the same notation for all scales with roughly the same g/p ratio (like meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant 495.88 / 1195.23 and garibaldi 498.12 / 1200.76).

meantone: 12&19, [<1, 2, 4, 7|, <0, -1, -4, -10|]
flattone: 19&26, [<1, 2, 4, -1|, <0, -1, -4, 9|]
dominant: 5&12, [<1, 2, 4, 2|, <0, -1, -4, 2|]
garibaldi: 12&29, [<1, 2, -1, -3|, <0, -1, 8, 14|]

>> To the mathematicians, the mapping from generators to prime number
>> intervals is everything. But you can't always obtain a unique
>> mapping by reverse-engineering the tuning. This was a very real
>> problem with claims made about the Zeng Bells a while back. More
>> than one mapping can result in the same tuning. Do we really want to
>> call them different temperaments if you can't hear or measure any
>> difference in the product? Maybe we do.
//
>This is an interesting point: the difference between "bug" and
>"superpelog" for instance is mainly in the mapping, and in the number of
>generators in a typical scale. "Superpelog" is basically a 14-note scale
>with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2
>3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is
>typical. But the generator / period ratio is very similar: 260.76 /
>1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and
>254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar
>tunings; the difference is in which notes are used to approximate
>consonant intervals. This is similar to the use of two different
>approximate fifths in ET's like 23-ET or 64-ET.

Erv Wilson has even suggested that the meantone and schismic
mappings may be separate in 12-tET, to the point of being able
to hear the difference.

Gene may have something to say about using tunings to categorize
temperaments -- he tried it initially but lately has been headed
toward using comma sequences. Gene, can you refresh us on the
gotchas of the tunings-based approach?

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> Another way to categorize tunings is by putting them on a branch of the
> scale tree (by their generator / period ratio if the period is
around an
> octave).

To actually use this to categorize you'd have to use a standard
version of the generator, such as the 1 < g < sqrt(period).

This would lump together scales that have different mappings,
> but sound roughly the same. For each specific generator / period ratio,
> a number of different mappings can be assigned depending on how far you
> continue the chain of generators in one direction or the other.

And also depending on what sort of numbers--for example, which prime
limit--we had in mind.

One
> thing this could be useful for is notation: it would make sense to use
> the same notation for all scales with roughly the same g/p ratio (like
> meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant
> 495.88 / 1195.23 and garibaldi 498.12 / 1200.76).

Where do you switch? Why, for instance, don't you continue on to
superpyth?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Gene may have something to say about using tunings to categorize
> temperaments -- he tried it initially but lately has been headed
> toward using comma sequences. Gene, can you refresh us on the
> gotchas of the tunings-based approach?

I was suggesting both of these in connection to family relationships.
Which 7-limit temperament, if any, should we simply call "meantone"?
Is there an 11-limit temperament we might simply call "meantone" also?
My suggestion was to sort it out, to the extent possible, excluding
anything too awful and then using TOP tuning.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> homomorphic mapping - I'd looked this up before, but I had to
> do it again. It looks like this phrase could be replaced with
> "homomorphism", since a morphism is already a type of mapping.
> And homomorphism is just a "general morphism".

Actually it's more like a morphism is a general homomorphism. Category
was (and occasionally still is) called "generalized abstract nonsense
theory" for a reason.

> finitely-generated subgroup of the positive rationals - I
> assume this allows for odd-limit and/or non-consective prime
> bases like {2, 3, 7 11}.

It doesn't distinguish the 7-limit from the 9-limit, but does allow
{2,3,7,11} and many other possibilities, including many hardly ever
even considered, much less adopted.

> ...to an...
>
> abstract free group of smaller rank - A group is a set that's
> closed under some operator, has an identity element, and a law
> of inverses. This was actually discussed in my 8th-grade
> Algebra class.

Ha! I thought they killed the New Math.

I'm not sure about the "free" part. It sounds
> like it's meant to rule out torsion.

It rules out torsion, it also says you can't keep subdividing, so it
rules out vector spaces.

http://en.wikipedia.org/wiki/Free_abelian_group

The "abstract" part
> looks superfluous to me.

I just wanted to make clear that tuning was not a part of the
definition at this point.

> kernel - I haven't a clue.

The kernel of a temperament is all of the intervals it sends to the
unison; these form another group.

>> abstract free group of smaller rank - A group is a set that's
>> closed under some operator, has an identity element, and a law
>> of inverses. This was actually discussed in my 8th-grade
>> Algebra class.
>
>Ha! I thought they killed the New Math.

My hillbilly school district couldn't afford to update their
textbooks. :)

This was actually the first chapter in the book, which I remember
as my favorite part of the class, but which all the other students
seemed to hate.

>> I'm not sure about the "free" part. It sounds
>> like it's meant to rule out torsion.
>
>It rules out torsion, it also says you can't keep subdividing, so it
>rules out vector spaces.
>
> http://en.wikipedia.org/wiki/Free_abelian_group

It looks like this "free" business is behind the "fundamental
theorem of arithmetic".

>> kernel - I haven't a clue.
>
>The kernel of a temperament is all of the intervals it sends to the
>unison; these form another group.

Aha! This explains why I've never seen a kernel written out.

For some reason I missed it here:
http://mathworld.wolfram.com/GroupKernel.html

-Carl

Hi Carl,

>> I would say a regular temperament is uniquely determined by a
>> homomorphic mapping from the p-limit, or possibly another
>> finitely-generated subgroup of the positive rationals, to an
>> abstract free group of smaller rank. This can be specified by
>> giving a wedgie, a kernel, or an explicit mapping.
>
>To many people on this list, like me, who haven't recently used or
>studied abstract algebra, category theory, geometric algebra, and
>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as
>meaningful as this:
>

This really isn't so mysterious at all if one is accustomed
to the mathematical terms that Gene is using here.

An abstract free group of smaller rank just
is an algebraic way of talking about
a lattice with fewer lattice directions.

That's by a standard result that a
finite free algebraic group is isomorphic to
Z^n (in algebraic lanaguage).

So wherever you see abstract free group
you can substitute the word lattice,
with no loss of understanding
(though an algebraicist might complain that
ones understanding of what is happening
isn't sufficiently abstract :-) ).
Its rank just means the dimension of
the lattice.

So a more geometrical way of saying the same
thing is:

A regular temperament is uniquely determined
by an interval addition preserving mapping
of a p-limit lattice to one of lower
dimension.

By interval addition preserving, this means that
e.g. if 5/4 * 6/5 = 3/2 and you temper it to
the pythagorean scale, you want to map it
to intervals such as 81/64, 32/27 and 3/2
with (81/64) * (32/27) = 3/2

Then for the rest of what it says:

You can either give the mapping explicitly,
or you can say which of the elements
map to the unison vector (this is what
is meant by the "kernel")
- which can then be extended to give a mapping
of the entire lattice

- or you can use a wedge product
- now that last bit I don't understand
quite yet, how he does these mappings
using wedge products, I seem to be missing
something. But the rest is clear.

I don't understand why he hasn't mentioned
what seems the most obvious way to do
it - that you just need to give a mapping
of the bases of the lattice
- if you say where 2, 3, and 5 map to
then you can say where every interval
maps to.

So for instance if 5/4 goes to 81/64
then 5/1 goes to 81/16
3/1 then just goes to 3/1
and if you require addition of intervals
to be preserved, then that
specifies an entire mapping
of the five limit lattice to the three
limit one - there is no room for further
choice having set out those requirements
on the mapping. A map that is defined
over the entire lattice must be defined
on its basis vectores 2, 3, 5 etc
so this procedure can always be followed
- unless I'm missing something.

At any rate I'm pretty sure that is what
it is saying - correct me if I'm wrong Gene!
And I'd be interested in a newbie explanation
of how wedge products are used to make
these mappings and to study them as that
seems to be one thing I'm missing right
now in my understanding of the
encyclopedia. It is probably some simple thing
like not understanding that vals
were being used to count the number of scale
degrees spanned by an interval.

Robert

Hi Robert,

>Hi Carl,
>
>>> I would say a regular temperament is uniquely determined by a
>>> homomorphic mapping from the p-limit, or possibly another
>>> finitely-generated subgroup of the positive rationals, to an
>>> abstract free group of smaller rank. This can be specified by
>>> giving a wedgie, a kernel, or an explicit mapping.
>>
>>To many people on this list, like me, who haven't recently used or
>>studied abstract algebra, category theory, geometric algebra, and
>>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as
>>meaningful as this:
>
>This really isn't so mysterious at all if one is accustomed
>to the mathematical terms that Gene is using here.

It was actually Dave who wrote that, though I found your
comments helpful.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> This is an interesting point: the difference between "bug" and
> "superpelog" for instance is mainly in the mapping, and in the number of
> generators in a typical scale. "Superpelog" is basically a 14-note scale
> with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2
> 3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is
> typical. But the generator / period ratio is very similar: 260.76 /
> 1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and
> 254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar
> tunings; the difference is in which notes are used to approximate
> consonant intervals. This is similar to the use of two different
> approximate fifths in ET's like 23-ET or 64-ET.

Wow; I was asking Paul Erlich about this sort of thing, mostly regarding a way that
you could make an sort of "atlas" of temperaments based on generator size. Clearly
many of these things defined precisely (if not uniquely) by their mapping overlap
greatly. Today's obsession:

If these things aren't exclusive, with clear-cut borders between them, can there at
least be a range of reasonability? Take "7-limit meantone", for example (my GOD!
can't get by a day without mention of meantone! you're all obsessed! :-). I started out
wanting to keep 7/4 greater than 3/2 greater than 5/4. Reasonable? My supreme
7th-grade math skills yeild a range of 5/9oct to 2/3oct. But that looks a little crazy
because if you pretend 9-equal is meantone, yer major third is smaller than yer minor
third!

So I tried keeping the fifth so that 5/4 > 6/5 > 7/6, and that seems to limit the fifth
to 4/7oct < g < 7/12oct. At one end, major and minor third become neutral, whilst
at the other, minor and subminor third are squinched into one. I don't see how these
can be objectively extended to other limits or how they could be calculated for TOP
tunings.

My rationale: if some dude prefers "bug" over "superpelog," what's to stop him from
using grossly-warped-yet-still-somehow-functional meantone?

> Another way to categorize tunings is by putting them on a branch of the
> scale tree (by their generator / period ratio if the period is around an
> octave). This would lump together scales that have different mappings,
> but sound roughly the same.

Herman I like this approach but still don't get this "scale tree." I tried to grow one of
my own but fell off and broke my wrist. Can anyone do a ground-up explanation? Or
an Encyclopaedia article?

Godspeed,
Jacob

on 8/15/04 7:22 PM, Dave Keenan <d.keenan@bigpond.net.au> wrote:

>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>
>>> Maybe I can start with a question. What uniquely determines a
>> temperament?
>
> This is an excellent question.
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>> I would say a regular temperament is uniquely determined by a
>> homomorphic mapping from the p-limit, or possibly another
>> finitely-generated subgroup of the positive rationals, to an
> abstract
>> free group of smaller rank. This can be specified by giving a
> wedgie,
>> a kernel, or an explicit mapping.

> But something we haven't agreed on is the following recasting of
> Kurt's question: "How can we tell, by listening or measuring,
> whether two tunings are different temperaments, or are merely
> slightly different tunings of the same temperament?"

Actually, yes this is quite a recasting. Because in fact my original
question was geared towards understanding the more technical sense of the
word "temperament" as it is often used on this list by Gene and others.

But I appreciate Dave's attempt here, and from its apparent tone I took it
as a rather playful exploration and inquiry into the possibility of a more
physical approach, rather than the definitive statement that Gene seemed to
hear. But Dave kind of asked for it with his Gene-mumbo-jumbo paragraph!

And yes, its true, I don't yet have what it takes to fully understand what
Gene said, so this set me back on my original quest a little. But I think
based on various clarifications that have come along, I can now fairly
safely recast Gene's answer to say that for most purposes on this list a
temperament is uniquely determined by the commas that are tempered out.
This is the confirmation that I was looking for. It confirms my
understanding of the way a temperament (as often used here and distinct from
the traditional common usage) is different from a tuning and may map to more
than one different tuning. I think I have heard this or something similar
echoed by several people now in this or other recent threads.

So if Gene ok's this, then I think maybe I can get back to my original
questioning that might shed more light on some of the gray areas that exist
for me along the polarity from temperament to tuning.

-Kurt

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> If these things aren't exclusive, with clear-cut borders between
them, can there at
> least be a range of reasonability? Take "7-limit meantone", for
example (my GOD!
> can't get by a day without mention of meantone! you're all obsessed!
:-). I started out
> wanting to keep 7/4 greater than 3/2 greater than 5/4. Reasonable?
My supreme
> 7th-grade math skills yeild a range of 5/9oct to 2/3oct. But that
looks a little crazy
> because if you pretend 9-equal is meantone, yer major third is
smaller than yer minor
> third!
>
> So I tried keeping the fifth so that 5/4 > 6/5 > 7/6, and that seems
to limit the fifth
> to 4/7oct < g < 7/12oct. At one end, major and minor third become
neutral, whilst
> at the other, minor and subminor third are squinched into one.

Sounds reasonable to me. You can relate this to the chroma thread:
25/24 is a chroma for a 7-note meantone MOS, and adopting it as a
comma turns the meantone MOS into 7-equal and squashes 5/4 and 6/5
together; 36/35 (squishing 6/5 and 7/6 together) is a 12-note meantone
chroma and works the same way with 12-et.

I don't see how these
> can be objectively extended to other limits or how they could be
calculated for TOP
> tunings.

I'm not quite sure what the question is, but other meantone chromas
which identify two 7-limit consonances are out there. Aside from
squashing 7/6 and 6/5 together, 12 also squashes 7/5 and 10/7
together, giving 50/49. 19 squashes 7/6 and 8/7 together; if you
regarded that as illegitimate you might want to put the boundries at
19 and 12. This has the advantage of giving an explicit criterion--you
are requiring that the meantone tuning not identify any two 7-limit
consonances nor put them in the wrong order, and this will be true
only if the tuning is somewhere between 19 and 12. This whole plan
pretty well breaks down when we get to the 11-limit, where 31, the
quintessential meantone tuning, identifies 11-limit consonances.

> Herman I like this approach but still don't get this "scale tree."
I tried to grow one of
> my own but fell off and broke my wrist. Can anyone do a ground-up
explanation? Or
> an Encyclopaedia article?

I think the reference was to the Stern-Brocot tree, which is closely
related to the Farey sequence.

http://mathworld.wolfram.com/Stern-BrocotTree.html

http://mathworld.wolfram.com/FareySequence.html

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > Gene may have something to say about using tunings to categorize
> > temperaments -- he tried it initially but lately has been headed
> > toward using comma sequences. Gene, can you refresh us on the
> > gotchas of the tunings-based approach?
>
> I was suggesting both of these in connection to family
relationships.
> Which 7-limit temperament, if any, should we simply
> call "meantone"?

none. the meantone family was clearly designed to
represent 5-limit harmony, and while some of the
intervals of various versions of meantone resemble
7-limit or higher-limit ratios, it was rarely
consciously used that way historically, the only
real exception being the enthusiasm of composers
to use "augmented-6th" chords (which resemble 4:5:6:7
tetrads closely in some meantones).

> Is there an 11-limit temperament we might simply call
> "meantone" also?

again, i think it's not a good idea.

BTW ... Paul pointed out to me that Huygens only
wrote about meantone as representing up to the 7-limit.
his name should be associated with one of those
rather than with the 11-limit version which bears
his name now.

> My suggestion was to sort it out, to the extent possible,
> excluding anything too awful and then using TOP tuning.

you mean "using TOP" to categorize? can you elaborate?

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
But I think
> based on various clarifications that have come along, I can now fairly
> safely recast Gene's answer to say that for most purposes on this list a
> temperament is uniquely determined by the commas that are tempered out.

This is precisely what I meant, though I restricted that to regular
temperaments. Actually, I thought it was also what I said.

Irregular temperaments are tempered *scales*, and so are different
beasties anyway.

hi Jacob,

i know you're talking about 7-limit meantone here,
but i thought i'd toss this in about 5-limit:

the limits of ET 5-limit meantones are generally
considered to be 12-ET at one extreme and 19-ET
at the other.

the main reason for this is the size of the two
different semitones which occur in all meantones
except 12-ET: the diatonic (as A:Bb) and chromatic
(as A:A#). in 12-ET the difference disappears and
these are actually the same.

in all other meantones the generator is smaller
than 2^(7/12) = 700 cents, and the chromatic semitone
is smaller than the diatonic.

narrowing the generator all the way down to
2^(11/19) = 694 + 14/19 cents, results in an
exact division of the 19edo "whole-tone"
(189 + 9/19 cents) into 3 parts. thus, the
diatonic semitone is exactly twice as large
as the chromatic semitone.

narrowing the generator to a size smaller than
2^(11/19) results in tunings which seem strange
for music composed with meantone in mind, because
the enharmonic difference (between, say, A# and Bb)
becomes larger than the difference between a
nominal and its accidentaled relative.

in the other direction, extending the generator
to a size larger than 2^(7/12) makes the chromatic
semitones larger than the diatonic, which resembles
pythagorean tuning, and becomes nearly identical
to pythagorean at 2^(31/53) = 701 + 47/53 cents.

-monz

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> Wow; I was asking Paul Erlich about this sort of thing,
> mostly regarding a way that you could make an sort of
> "atlas" of temperaments based on generator size. Clearly
> many of these things defined precisely (if not uniquely)
> by their mapping overlap greatly. Today's obsession:
>
> If these things aren't exclusive, with clear-cut borders
> between them, can there at least be a range of reasonability?
> Take "7-limit meantone", for example (my GOD! can't get by
> a day without mention of meantone! you're all obsessed! :-).

> I started out wanting to keep 7/4 greater than 3/2 greater
> than 5/4. Reasonable? My supreme 7th-grade math skills
> yeild a range of 5/9oct to 2/3oct. But that looks a
> little crazy because if you pretend 9-equal is meantone,
> yer major third is smaller than yer minor third!
>
> <etc. -- snip>

oops ... i had meant to include this, which makes
it all plain in a very visual way:

http://tonalsoft.com/enc/index2.htm?../monzo/meantone/cycles.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Jacob,
>
>
> i know you're talking about 7-limit meantone here,
> but i thought i'd toss this in about 5-limit:
>
>
> the limits of ET 5-limit meantones are generally
> considered to be 12-ET at one extreme and 19-ET
> at the other.
>
>
> the main reason for this is the size of the two
> different semitones which occur in all meantones
> except 12-ET: the diatonic (as A:Bb) and chromatic
> (as A:A#). in 12-ET the difference disappears and
> these are actually the same.
>
>
> in all other meantones the generator is smaller
> than 2^(7/12) = 700 cents, and the chromatic semitone
> is smaller than the diatonic.
>
>
> narrowing the generator all the way down to
> 2^(11/19) = 694 + 14/19 cents, results in an
> exact division of the 19edo "whole-tone"
> (189 + 9/19 cents) into 3 parts. thus, the
> diatonic semitone is exactly twice as large
> as the chromatic semitone.
>
>
> narrowing the generator to a size smaller than
> 2^(11/19) results in tunings which seem strange
> for music composed with meantone in mind, because
> the enharmonic difference (between, say, A# and Bb)
> becomes larger than the difference between a
> nominal and its accidentaled relative.
>
>
> in the other direction, extending the generator
> to a size larger than 2^(7/12) makes the chromatic
> semitones larger than the diatonic, which resembles
> pythagorean tuning, and becomes nearly identical
> to pythagorean at 2^(31/53) = 701 + 47/53 cents.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> none. the meantone family was clearly designed to
> represent 5-limit harmony, and while some of the
> intervals of various versions of meantone resemble
> 7-limit or higher-limit ratios, it was rarely
> consciously used that way historically, the only
> real exception being the enthusiasm of composers
> to use "augmented-6th" chords (which resemble 4:5:6:7
> tetrads closely in some meantones).

On the one hand we have Dave, who thinks it all devolves to the actual
tuning. On the other hand we have you, who think that the precise same
tuning, and a consistent tuning map, doesn't mean diddly. This is
using history as a hindrence rather than a help, isn't it?

For certain, we can't very well adopt both points of view!

> BTW ... Paul pointed out to me that Huygens only
> wrote about meantone as representing up to the 7-limit.
> his name should be associated with one of those
> rather than with the 11-limit version which bears
> his name now.

If we are going to be picky, he wrote about 31-equal.

> you mean "using TOP" to categorize? can you elaborate?

7-limit meantone, which you say isn't meantone, has the exact same TOP
tuning as 5-limit meantone. So we have two temperaments, which have
exactly the same tuning and are being used in an entirely consistent
way, but giving them the same name brings down the wrath of the tuning
Inquisition. Let us by all means not call them by the same name, so as
to keep the fact that they have the same tuning hidden.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I can now fairly safely recast Gene's answer to say that
> for most purposes on this list a temperament is uniquely
> determined by the commas that are tempered out. This is
> the confirmation that I was looking for. It confirms my
> understanding of the way a temperament (as often used here
> and distinct from the traditional common usage) is different
> from a tuning and may map to more than one different tuning.
> I think I have heard this or something similar echoed by
> several people now in this or other recent threads.

i haven't really been sure where you going with this,
but here's my 2 cents:

it seems to me you're talking about the way certain EDOs,
for example, can be members of many different temperament
families, i.e.:

- 12edo belongs to meantone, schismic, augmented (diesic), etc.,
- 19edo belongs to meantone, magic, and kleismic
- 31edo belongs to meantone, miracle

some of these classifications can be seen in the very
bottom applet on my "bingo" page:

http://tonalsoft.com/enc/index2.htm?bingo.htm

is that what you're getting at?

-monz

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > none. the meantone family was clearly designed to
> > represent 5-limit harmony, and while some of the
> > intervals of various versions of meantone resemble
> > 7-limit or higher-limit ratios, it was rarely
> > consciously used that way historically, the only
> > real exception being the enthusiasm of composers
> > to use "augmented-6th" chords (which resemble 4:5:6:7
> > tetrads closely in some meantones).
>
> On the one hand we have Dave, who thinks it all devolves
> to the actual tuning. On the other hand we have you, who
> think that the precise same tuning, and a consistent tuning
> map, doesn't mean diddly. This is using history as a
> hindrence rather than a help, isn't it?
>
> For certain, we can't very well adopt both points of view!
>
> > BTW ... Paul pointed out to me that Huygens only
> > wrote about meantone as representing up to the 7-limit.
> > his name should be associated with one of those
> > rather than with the 11-limit version which bears
> > his name now.
>
> If we are going to be picky, he wrote about 31-equal.

if we're really going to be picky, he wrote about
how 31-equal comes so close to 1/4-comma meantone.

in fact, it's very interesting to me to see Huygens
refer to 31edo as "Division of the Octave into 31
Equal Parts", but to call 1/4-comma meantone
"temperament ordinaire" [ordinary temperament] or
often simply just plain "temperament".

> > you mean "using TOP" to categorize? can you elaborate?
>
> 7-limit meantone, which you say isn't meantone, has the
> exact same TOP tuning as 5-limit meantone. So we have two
> temperaments, which have exactly the same tuning and are
> being used in an entirely consistent way, but giving them
> the same name brings down the wrath of the tuning
> Inquisition. Let us by all means not call them by the
> same name, so as to keep the fact that they have the same
> tuning hidden.

wow, quit snarling for a while, would you? i merely
gave an opinion which you solicited and now all of a
sudden you mention the "tuning Inquisition" with what
sounds to me like a clear reference to me.

i think your wrath is simply the result of misunderstanding
what i meant, which i didn't express clearly.

of course, these tunings all belong to the meantone
family ... i never said that in my post, so it's clear
why you didn't get that from what i wrote.

i just meant that i think plain-old "meantone" should
be reserved for 5-limit, and then "septimal meantone"
or "huygens" for 7-limit, and whatever appropriate name
for 11-limit, etc.

certainly, i want to see Aunt Martha, Illegitimate
Uncle Simon, and all the other long-lost cousins
of the (perhaps somewhat dysfunctional?) meantone family.

... i can't wait to see what happens when they
get into arguments! ... hmm ... what a great
idea for a polymicrotonal opera ...

-monz

on 8/16/04 2:02 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> But I think
>> based on various clarifications that have come along, I can now fairly
>> safely recast Gene's answer to say that for most purposes on this list a
>> temperament is uniquely determined by the commas that are tempered out.
>
> This is precisely what I meant, though I restricted that to regular
> temperaments. Actually, I thought it was also what I said.

I'll reply to this in the original thread shortly.

> Irregular temperaments are tempered *scales*, and so are different
> beasties anyway.

So is what you are calling "tempered scales" closer to the historical
meaning of "temperament"?

Prior to physics-based tuning theory, everyone was talking about scales,
right? At that time people didn't know about frequency ratios per-se,
although length-of-string ratios are much older. On the other hand there
was clearly some sense even then that exact intervals were being tempered,
in spite of the lack of physical theory defining exact intervals.
Nonetheless it all dealt with application to scales and tunings.

My sense is that "temperament" and "tempered scale" used to be synonymous,
though I'm not quite sure when the term "temperament" was invented. I'm
presuming it was well before "well-temperament".

Regardless of history: what exactly is your distinction between
"temperament" and "tempered scale" and how does a "tempered scale" relate to
a tuning? Is this a generally agreed distinction, a distinction unique to
this community, or a more universal one?

-Kurt

>Prior to physics-based tuning theory, everyone was talking
>about scales, right?

Temperaments have been abstract entities apart from scales
since I joined this list in 1997, though Gene made the
distinction more precise and has stressed its importance.
I didn't fully grok it until Gene came along.

>Regardless of history: what exactly is your distinction between
>"temperament" and "tempered scale" and how does a "tempered scale"
>relate to a tuning? Is this a generally agreed distinction, a
>distinction unique to this community, or a more universal one?

I'll let Gene answer this, but note that there is also such a
thing as a scale (apart from a tempered scale) and it's importance
is something I have stressed in our discussions. But I can't find
Gene's definition just now, in the Encyclopaedia or on his site.

-Carl

on 8/16/04 2:02 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> But I think
>> based on various clarifications that have come along, I can now fairly
>> safely recast Gene's answer to say that for most purposes on this list a
>> temperament is uniquely determined by the commas that are tempered out.
>
> This is precisely what I meant, though I restricted that to regular
> temperaments. Actually, I thought it was also what I said.

Yes, basically but you gave more options including an explicit mapping, and
you used the term "kernel" instead of "comma".

But in any case I can return to my original line of questioning now, which
should clarify to Monz what I was up to. Actually I've already learned so
much that returning to my original line of questions feels slightly
artificial, but I'll try to do it anyway since otherwise I've kept all the
intervening thoughts to myself.

Lets look at starting with the 5-limit and tempering out the 225/224 to make
the 225:128 into an exact 7:4. So my understanding is no matter how you get
rid of the 225/224 comma starting from the 5-limit, you have the same
temperament as a result.

So let's look at at making a tuning in this temperament. One approach that
comes to mind is to alter the 3:2 an the 5:4 by the same factor in order to
make the 7:4 exact. So this involves applying the ratio (225/224)^(1/4) to
3:2 and to 5:4. This value (225/224)^(1/4) is something like a comma
itself, though I guess it isn't standard use of the term. But this is one
of those irrational comma-like quantities that I had talked about. It is a
distributed granule of a comma. Now I understand this is not called a
comma. Originally I thought of tempering as bending the lattice in various
ways and I thought of the ratios involved in the bending as being some kind
of comma. And I thought of the different ways of altering the lattice in
order to achieve the different tempered tunings as different temperaments,
which were to me *structurally* different even though the same comma was
being tempered out. This might be clearer if I give an example of a
different tempered tuning from the same "temperament" that to me is
structurally different.

So let's look at another example. One thought about tempering out 225/224
is that if 4:7 is to be made exact, why not make 5:7 or 6:7 exact too. One
or the other has to be picked; you can't have both by tweaking a lattice
interval in all occurrences (which is what I'm looking at for the moment).
So let's say we want to make 5:7 exact at the expense of 6:7. This allows
you to have an exact 4:5:7 chord. Now this means *not* tweaking the 5:4
lattice interval, but instead tweaking only the 3:2, by the square of the
ratio that was used in the previous example. Now to me this is structurally
different because it involves a different assumption of what the goal of the
tempering process is. Does that make sense? Yet the result is considered
the same temperament. Yet I guess Gene would say this is all working in a
real number space rather than an abelian group, and so it is just a matter
or tuning and not of defining a temperament.

So this pretty much finishes my original line of thinking (or the part of it
that I remember) which I think reveals most of the confusion I was having
about "what is a temperament". At this moment I have no specific questions
but will be interested in any responses to the above lines of thinking. I
hope I stated it clearly enough.

-Kurt

Hi Kurt (and Gene)!

Sorry to keep butting in here, just a few points:

>Yes, basically but you gave more options including an explicit
>mapping, and you used the term "kernel" instead of "comma".

A kernel is actually the set of all the commas that vanish
in a temperament -- which I think also includes things like
powers of vanishing commas, and therefore contains an infinite
number of commas.

>Lets look at starting with the 5-limit and tempering out the
>225/224 to make the 225:128 into an exact 7:4. So my
>understanding is no matter how you get rid of the 225/224
>comma starting from the 5-limit, you have the same temperament
>as a result.

Strictly speaking, 225:224 is a 7-limit interval, so you
can't temper it out of the 5-limit lattice.

>So let's look at at making a tuning in this temperament.

Just to refresh, a tuning is an infinite thing. Temperaments
are tunings, but scales are not. Sometimes scales can be
explained in terms of tunings, though, as Gene recently did
in his "Reverse engineering a scale" thread.

>One approach that comes to mind is to alter the 3:2 an the
>5:4 by the same factor in order to make the 7:4 exact. So
>this involves applying the ratio (225/224)^(1/4) to 3:2 and
>to 5:4. This value (225/224)^(1/4) is something like a comma
>itself, though I guess it isn't standard use of the term.
>But this is one of those irrational comma-like quantities
>that I had talked about.

It's an "error". That's what we'd call it. Sorry I couldn't
think of that the other night!

...That's all for now. Let me hint again that your "original
line of questioning" is getting you close to rediscovering TOP
tuning. OTOH, we were close for years around here until Paul
put it all together. So if you want to cut to the chase, I
can come over and explain it sometime.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> I'll let Gene answer this, but note that there is also such a
> thing as a scale (apart from a tempered scale) and it's importance
> is something I have stressed in our discussions. But I can't find
> Gene's definition just now, in the Encyclopaedia or on his site.

That definition was much hated. What about this:

A discrete set of real numbers including 0 giving the value, in cents,
of frequency ratios relative to the base frequency represented by 0. A
periodic scale has an indexing mapping s from the integers and a
positive integer n such that s[i+n]-s[i] is a constant for all i. A
finite scale is a finite set of cents values, including 0. Scales can
also be given multipliciatively, including using rational numbers
only, so long as the corresponding values in cents satisfy the
requirements for being a scale.

Note that using this definition, the p-limit for any odd prime p is
not a scale, but any equal division of the octave will be.

>> I'll let Gene answer this, but note that there is also such a
>> thing as a scale (apart from a tempered scale) and it's importance
>> is something I have stressed in our discussions. But I can't find
>> Gene's definition just now, in the Encyclopaedia or on his site.
>
>That definition was much hated.

I remember. I wish I had it to look back on now. It's probably
here in my inbox. Any terms other than "scale" it might have
contained?

>What about this:
>
>A discrete set of real numbers including 0 giving the value, in
>cents, of frequency ratios relative to the base frequency
>represented by 0.

This sounds imprecise. In particular, it sounds like we are
being restricted to rational numbers.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> Actually, yes this is quite a recasting. Because in fact my
original
> question was geared towards understanding the more technical sense
of the
> word "temperament" as it is often used on this list by Gene and
others.
>
> But I appreciate Dave's attempt here, and from its apparent tone I
took it
> as a rather playful exploration and inquiry into the possibility
of a more
> physical approach, rather than the definitive statement that Gene
seemed to
> hear. But Dave kind of asked for it with his Gene-mumbo-jumbo
paragraph!
>

Yes I did, didn't I. :-)

I wanted to make the point, in a humorous way, that this isn't the
tuning-math list, and it was quite possible to say the same thing in
a way that you, and many others, would have immediately understood,
as others have subsequently done.

Then I wanted to remind us of the possibility of an extreme
experiential point of view as balance to Gene's extreme mathematical
one.

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >A discrete set of real numbers including 0 giving the value, in
> >cents, of frequency ratios relative to the base frequency
> >represented by 0.
>
> This sounds imprecise. In particular, it sounds like we are
> being restricted to rational numbers.

Only if you think a ratio has to be a ratio of integers; but this is
not the case. The opposite viewpoint goes back to ancient Greece and
has been accepted ever since; if you say the ratio of the side of a
square to its diagonal cannot be a ratio of integers, which the Greeks
were willing to say, you admit ratios need not be commensureable; as
Euclid puts it "two magnitudes are commensurable if and only if their
ratio is the ratio of a number to a number", where of course for
"magnitude" we would say "positive real number" and for "number" we
would say "positive integer".

Now, probably we don't want to refer people to Euclid in order to sort
this all out, so what would be good at this point is a suggestion for
saying it in a way which would be less confusing.

on 8/16/04 5:48 PM, Carl Lumma <ekin@lumma.org> wrote:

>
> Hi Kurt (and Gene)!
>
> Sorry to keep butting in here, just a few points:
>
>> Yes, basically but you gave more options including an explicit
>> mapping, and you used the term "kernel" instead of "comma".
>
> A kernel is actually the set of all the commas that vanish
> in a temperament -- which I think also includes things like
> powers of vanishing commas, and therefore contains an infinite
> number of commas.

Yes, quite frankly its all the same to me. Maybe the kernel then is the
"expansion" of the commas that are involved into an entire *mapping*? Just
taking a guess. It sounded like like a mapping makes everything explicit
and doesn't require a list of commas tempered out, and is a more general
entity. Yet Gene didn't equate kernel to mapping, so mapbe kernel is still
less general.

>> Lets look at starting with the 5-limit and tempering out the
>> 225/224 to make the 225:128 into an exact 7:4. So my
>> understanding is no matter how you get rid of the 225/224
>> comma starting from the 5-limit, you have the same temperament
>> as a result.
>
> Strictly speaking, 225:224 is a 7-limit interval, so you
> can't temper it out of the 5-limit lattice.

Yes, yes, I keep forgetting. The way I naturally think keeps getting in the
way. I think: "Ok, I'm in the 5-limit. Gee here's a good approximation to
4:7. Let's make it exact." So slopily it looked to me like I was tempering
the 5 limit. And damn it, I *was* tempering the 5 limit. You math heads
get out of here. You're obscuring the obvious. ;) See why math is a
problem? It makes us retroactively unconscious. ;)

>> So let's look at at making a tuning in this temperament.
>
> Just to refresh, a tuning is an infinite thing.

Ok, I was using tuning as a finite thing, because damn it if you try to tune
an infinite thing you won't finish tuning it before you die. You math heads
don't know what "tuning" really means, now do you. Get off your computers
and touch something real. ;)

> Temperaments
> are tunings, but scales are not. Sometimes scales can be
> explained in terms of tunings, though, as Gene recently did
> in his "Reverse engineering a scale" thread.
>
>> One approach that comes to mind is to alter the 3:2 an the
>> 5:4 by the same factor in order to make the 7:4 exact. So
>> this involves applying the ratio (225/224)^(1/4) to 3:2 and
>> to 5:4. This value (225/224)^(1/4) is something like a comma
>> itself, though I guess it isn't standard use of the term.
>> But this is one of those irrational comma-like quantities
>> that I had talked about.
>
> It's an "error". That's what we'd call it. Sorry I couldn't
> think of that the other night!

I think you might have, it sounds really familiar. But I didn't like the
word and don't like it now. I know what you mean, but to me I was
correcting an error in the 4:7, so that was the distributed correction.

I can learn to think like you and I have already assimilated a lot of
abstract stuff that now causes me to have problems communicating with
ordinary people. Really! ;)

> ...That's all for now. Let me hint again that your "original
> line of questioning" is getting you close to rediscovering TOP
> tuning. OTOH, we were close for years around here until Paul
> put it all together. So if you want to cut to the chase, I
> can come over and explain it sometime.

Sure. My intuition is telling me that TOP becomes a way of classifying
tempered scales.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So let's look at at making a tuning in this temperament. One
approach that
> comes to mind is to alter the 3:2 an the 5:4 by the same factor in
order to
> make the 7:4 exact. So this involves applying the ratio
(225/224)^(1/4) to
> 3:2 and to 5:4. This value (225/224)^(1/4) is something like a comma
> itself, though I guess it isn't standard use of the term.

Strange you should mention this; I just posted something for Monz over
on tuning-math which mentions this tuning (which I call 1/4-kleismic,
in analogy to 1/4-comma.)

Now to me this is structurally
> different because it involves a different assumption of what the
goal of the
> tempering process is. Does that make sense?

It's the reverse of what I would mean by "structurally different";
we've got exactly the same structure, and variations in tuning which
are, after all, not that great.

Yet the result is considered
> the same temperament. Yet I guess Gene would say this is all
working in a
> real number space rather than an abelian group, and so it is just a
matter
> or tuning and not of defining a temperament.

I would say it is working with the tuning map and not the temperament
map. Tuning "factors through" the temperament, as mathematicians are
wont to say (before they start drawing diagrams with lots of arrows,
so I'd better stop now.)

>> A kernel is actually the set of all the commas that vanish
>> in a temperament -- which I think also includes things like
>> powers of vanishing commas, and therefore contains an infinite
>> number of commas.
>
>Yes, quite frankly its all the same to me. Maybe the kernel then
>is the "expansion" of the commas that are involved into an entire
>*mapping*? Just taking a guess. It sounded like like a mapping
>makes everything explicit and doesn't require a list of commas
>tempered out, and is a more general entity. Yet Gene didn't equate
>kernel to mapping, so mapbe kernel is still less general.

Well, I don't know much about kernels. Just learned about them
the other day. I found the mathworld definition helpful. Around
here, the word has usually occurred in a phrase like: 'Is this
interval in the tuning's kernel?' This phrase just means: 'Does
this interval vanish in the tuning?'

>>> Lets look at starting with the 5-limit and tempering out the
>>> 225/224 to make the 225:128 into an exact 7:4. So my
>>> understanding is no matter how you get rid of the 225/224
>>> comma starting from the 5-limit, you have the same temperament
>>> as a result.
>>
>> Strictly speaking, 225:224 is a 7-limit interval, so you
>> can't temper it out of the 5-limit lattice.
>
>Yes, yes, I keep forgetting. The way I naturally think keeps
>getting in the way. I think: "Ok, I'm in the 5-limit. Gee
>here's a good approximation to 4:7. Let's make it exact." So
>slopily it looked to me like I was tempering the 5 limit. And
>damn it, I *was* tempering the 5 limit.

You're tempering 5-limit intervals, but saying 'let's make it
an exact 4:7' implies a 7-limit universe.

>You math heads get out of here. You're obscuring the obvious. ;)
>See why math is a problem? It makes us retroactively unconscious. ;)

Well I never! You're the one who taught me about integrals!

>>> So let's look at at making a tuning in this temperament.
>>
>> Just to refresh, a tuning is an infinite thing.
>
>Ok, I was using tuning as a finite thing, because damn it if you
>try to tune an infinite thing you won't finish tuning it before
>you die. You math heads don't know what "tuning" really means,
>now do you. Get off your computers and touch something real. ;)

I've been bitten by using the word "temperament" to mean scale
several times in the last year or so. In one such case I
retorted that infinite things don't matter in music theory. But
in fact they do.

>> Temperaments
>> are tunings, but scales are not. Sometimes scales can be
>> explained in terms of tunings, though, as Gene recently did
>> in his "Reverse engineering a scale" thread.
>>
>>> One approach that comes to mind is to alter the 3:2 an the
>>> 5:4 by the same factor in order to make the 7:4 exact. So
>>> this involves applying the ratio (225/224)^(1/4) to 3:2 and
>>> to 5:4. This value (225/224)^(1/4) is something like a comma
>>> itself, though I guess it isn't standard use of the term.
>>> But this is one of those irrational comma-like quantities
>>> that I had talked about.
>>
>> It's an "error". That's what we'd call it. Sorry I couldn't
>> think of that the other night!
>
>I think you might have, it sounds really familiar. But I didn't
>like the word and don't like it now. I know what you mean, but
>to me I was correcting an error in the 4:7, so that was the
>distributed correction.

From the JI point of view, there is no error in 4:7. You're
putting error into the 3 and/or 5 axis to get 225/128 = 7/4.

>> ...That's all for now. Let me hint again that your "original
>> line of questioning" is getting you close to rediscovering TOP
>> tuning. OTOH, we were close for years around here until Paul
>> put it all together. So if you want to cut to the chase, I
>> can come over and explain it sometime.
>
>Sure. My intuition is telling me that TOP becomes a way of
>classifying tempered scales.

Well, Gene suggested it be used to define the canonical tuning
for any temperament, and this idea seems to have met with a
cautious quasi-consensus, which is the best kind of consensus
ever likely to emerge from the tuning-math list. :)

Once you have a way of finding a canonical tuning for any
temperament, you can try to put temperaments into families
based on the tunings (and ultimately scales) they give rise
to. Another approach is to base the families on comma
sequences. I'm not clear on the pros/cons of these approaches.

-Carl

>> Yet the result is considered
>> the same temperament. Yet I guess Gene would say this is all
>> working in a real number space rather than an abelian group,
>> and so it is just a matter or tuning and not of defining a
>> temperament.
>
>I would say it is working with the tuning map and not the temperament
>map. Tuning "factors through" the temperament, as mathematicians are
>wont to say (before they start drawing diagrams with lots of arrows,
>so I'd better stop now.)

Another way of saying it is that you choose a mapping (temperament)
and an error function, calculate a tuning by minimizing the error
given by your function and mapping, and finally produce a scale by
choosing a reasonable number of pitches from that tuning.

Does this help?

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So let's say we want to make 5:7 exact at the expense of 6:7. This
allows
> you to have an exact 4:5:7 chord. Now this means *not* tweaking the 5:4
> lattice interval, but instead tweaking only the 3:2, by the square
of the
> ratio that was used in the previous example.

In other words, we flatten the fifth by sqrt(225/224), and leave 2, 5,
and 7 pure. This gives us a fifth we can use as a meantone fifth--in
fact, something pretty close to the fifth of 55-equal, which should
make Monz happy. It is 2*sqrt(14)/5, or 698.099 cents in size, and it
gives us a meantone third sharp by 3136/3125, or 6.083 cents.
Meanwhile, we also have a pure third; we've got an inconsistent system
cooking here, with two kinds of major third. Find an intelligent way
of putting together two or more chains of 1/2-kleismic meantone fifths
separated by 5/4, 7/4, or 7/5, and you're in business. A similar idea
would be to use the 1/2-kleismic major third as a magic temperament
generator.

on 8/16/04 7:27 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> Yet the result is considered
>>> the same temperament. Yet I guess Gene would say this is all
>>> working in a real number space rather than an abelian group,
>>> and so it is just a matter or tuning and not of defining a
>>> temperament.
>>
>> I would say it is working with the tuning map and not the temperament
>> map. Tuning "factors through" the temperament, as mathematicians are
>> wont to say (before they start drawing diagrams with lots of arrows,
>> so I'd better stop now.)
>
> Another way of saying it is that you choose a mapping (temperament)
> and an error function, calculate a tuning by minimizing the error
> given by your function and mapping, and finally produce a scale by
> choosing a reasonable number of pitches from that tuning.
>
> Does this help?

Yes, this brings it into another familiar territory since in fact I have
thought about and asked questions in the past about optimizing ...umm...
scales, I think. And in fact TOP and standard-octave TOP variants came up
at that time. This leaves each pitch free to be optimized independently.

In my examples I was giving it more "structure", i.e. tuning an entire
lattice at once, perhaps you would say by scaling it. I wanted to
distinguish that special-case from the general case of tuning each pitch
independently, but I can see how that is probably not important enough to
create specific mathematical structures for, because there are too many
choices available to be worth categorizing. As Gene said the various
tunings don't differ by much. In fact my special cases fit into the general
case simply by stating what specifically I want to optimize, e.g. the 4:7
and the 4:5 with no consideration of the 4:6. That's coming at it from the
other end though, and is not the most natural way for me to think about it.
I don't like applying generality and limiting it if instead I can apply less
generality. But that's just me, and probably not (for example) Paul, who
I'll bet likes to apply the most generality possible and then apply
constraints to it.

-Kurt

Gene Ward Smith wrote:

> I think the reference was to the Stern-Brocot tree, which is closely
> related to the Farey sequence.
> > http://mathworld.wolfram.com/Stern-BrocotTree.html
> > http://mathworld.wolfram.com/FareySequence.html

Specifically, the reference is to Erv Wilson's application of the Stern-Brocot tree for classifying the structures of what he calls "MOS" scales (which stands for "moments of symmetry"). The fractions on the scale tree refer to the relative size of the scale's generator; different sizes of generator result in different scale structures.

http://www.anaphoria.com/sctree.PDF
http://www.anaphoria.com/hrgm.PDF
http://www.anaphoria.com/line.PDF

These are all available on the Wilson Archives web site:

http://www.anaphoria.com/wilson.html

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>Another way to categorize tunings is by putting them on a branch of the >>scale tree (by their generator / period ratio if the period is
> > around an > >>octave).
> > > To actually use this to categorize you'd have to use a standard
> version of the generator, such as the 1 < g < sqrt(period).

Note that I wrote "tunings", not "temperaments". Categorizing tunings by their position on the scale tree associates tunings that have the same scale structure (in numbers of small and large steps). Any particular tuning, with a particular pattern of MOS scales that can be displayed on a horagram, is compatible with a number of different temperaments, although usually only a few make sense. Similarly, any particular temperament can be tuned in different ways, producing different scale structures. Both of these categorizations are useful in different ways.

Take for example Wilson's golden horagram #9, which is labeled "Hanson". This has a generator of (4 phi + 1) / (15 phi + 4), or approximately 317.17 cents. One mapping that makes sense for this tuning is [<1, 0, 1, -3|, <0, 6, 5, 22|], which is a temperament for which the name "Hanson" has been suggested (it's the one that best fits the 14/53 generator implied in Larry Hanson's paper). But for more accuracy you can use the alternative mapping [<1, 0, 1, 11|, <0, 6, 5, -31|]. Or if you can tolerate a larger error, [<1, 0, 1, 2|, <0, 6, 5, 3|] will do. But however you interpret it, it's the same tuning, with the same scale structure.

> One > >>thing this could be useful for is notation: it would make sense to use >>the same notation for all scales with roughly the same g/p ratio (like >>meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant >>495.88 / 1195.23 and garibaldi 498.12 / 1200.76).
> > > Where do you switch? Why, for instance, don't you continue on to
> superpyth?

Where you switch depends on how many notes you use in the notation. If you're using 12 notes, the node in the scale tree adjacent to 5/12 is 7/17; the useful range of superpyth is on the other side of 7/17, so it's on a different branch. If you carry the generators out far enough, you'd use different notations for a version of meantone with a 504.13 / 1201.7 ratio and a version of flattone with a 507.14 / 1202.54 ratio.

>> Another way of saying it is that you choose a mapping (temperament)
>> and an error function, calculate a tuning by minimizing the error
>> given by your function and mapping, and finally produce a scale by
>> choosing a reasonable number of pitches from that tuning.
>>
>> Does this help?
>
>Yes, this brings it into another familiar territory since in fact I have
>thought about and asked questions in the past about optimizing ...umm...
>scales, I think. And in fact TOP and standard-octave TOP variants came up
>at that time. This leaves each pitch free to be optimized independently.
>
>In my examples I was giving it more "structure", i.e. tuning an entire
>lattice at once, perhaps you would say by scaling it.

Actually, this is exactly what TOP does -- tunes the whole lattice
by 'scaling' it.

>I wanted to distinguish that special-case from the general case of
>tuning each pitch independently,

As far as regular temperaments are concerned, one doesn't get to
tune the pitches independently. Tuning pitches independently gives
you what Gene would call a circulating temperament.

>but I can see how that is probably not important enough to
>create specific mathematical structures for, because there are too many
>choices available to be worth categorizing. As Gene said the various
>tunings don't differ by much. In fact my special cases fit into the
>general case simply by stating what specifically I want to optimize,
>e.g. the 4:7 and the 4:5 with no consideration of the 4:6.

Now you just need to specify how many dimensions you want your
temperament to have (or equivalently which commas you want to
vanish) and you can have what you want!

-Carl

hi Kurt,

not really saying much here to clarify your questions,
but i know a bit about the historical background ...

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Prior to physics-based tuning theory, everyone was
> talking about scales, right? At that time people
> didn't know about frequency ratios per-se, although
> length-of-string ratios are much older. On the other
> hand there was clearly some sense even then that exact
> intervals were being tempered, in spite of the lack of
> physical theory defining exact intervals. Nonetheless
> it all dealt with application to scales and tunings.

regarding tuning by measurement of string-lengths:

the earliest literature which definitely describes tuning
by measurement of string-length ratios, AFAIK, is by
Philolaus c.400s BC, and then a bit later and in much
more detail by Archytas, c.300s BC ... both ancient Greeks.

see my so far still very crude timeline of music-theory:

http://tonalsoft.com/enc/index2.htm?../monzo/timeline/timeline.htm

however, studying a Babylonian tablet from c.1800 BC,
i believe that i have deciphered a Sumerian method
for computing "pythagorean" tuning c.3000-2500 BC:

http://tonalsoft.com/enc/index2.htm?../monzo/sumerian/sumerian-tuning.
htm

OR

http://tinyurl.com/5rusu

regarding temperament:

this is at least as old as ancient Greece, 300s BC.

Aristoxenus, perhaps the most profound of all the
ancient Greek music-theorists, steadfastly refused
to mention ratios at all in his treatise, preferring
instead to employ the new geometrical ideas that had
only recently been laid down by Euclid.

he discusses "tuning by concords" (successive
"perfect-4ths" and "perfect-5ths"), in such a way
that after finding 12 different notes, one still has
another "perfect-5th" between the two ends of the
chain, which thus makes it a circle and not a chain.

this cannot happen using acoustically "pure" pythagorean
tuning, and so it must necessarily indicate temperament.
see:

http://tonalsoft.com/enc/index2.htm?../monzo/aristoxenus/318tet.
htm&concords

OR

http://tinyurl.com/6ews8

and scroll down a bit further to see some Excel graphs i
made of another examination of his "tuning by concords".

-monz

hi Kurt,

not really saying much here to clarify your questions,
but i know a bit about the historical background ...

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Prior to physics-based tuning theory, everyone was
> talking about scales, right? At that time people
> didn't know about frequency ratios per-se, although
> length-of-string ratios are much older. On the other
> hand there was clearly some sense even then that exact
> intervals were being tempered, in spite of the lack of
> physical theory defining exact intervals. Nonetheless
> it all dealt with application to scales and tunings.

regarding tuning by measurement of string-lengths:

the earliest literature which definitely describes tuning
by measurement of string-length ratios, AFAIK, is by
Philolaus c.400s BC, and then a bit later and in much
more detail by Archytas, c.300s BC ... both ancient Greeks.

see my so far still very crude timeline of music-theory:

http://tonalsoft.com/enc/index2.htm?../monzo/timeline/timeline.htm

however, studying a Babylonian tablet from c.1800 BC,
i believe that i have deciphered a Sumerian method
for computing "pythagorean" tuning c.3000-2500 BC:

http://tonalsoft.com/enc/index2.htm?../monzo/sumerian/sumerian-tuning.
htm

OR

http://tinyurl.com/5rusu

regarding temperament:

this is at least as old as ancient Greece, 300s BC.

Aristoxenus, perhaps the most profound of all the
ancient Greek music-theorists, steadfastly refused
to mention ratios at all in his treatise, preferring
instead to employ the new geometrical ideas that had
only recently been laid down by Euclid.

he discusses "tuning by concords" (successive
"perfect-4ths" and "perfect-5ths"), in such a way
that after finding 12 different notes, one still has
another "perfect-5th" between the two ends of the
chain, which thus makes it a circle and not a chain.

this cannot happen using acoustically "pure" pythagorean
tuning, and so it must necessarily indicate temperament.
see:

http://tonalsoft.com/enc/index2.htm?../monzo/aristoxenus/318tet.
htm&concords

OR

http://tinyurl.com/6ews8

and scroll down a bit further to see some Excel graphs i
made of another examination of his "tuning by concords".

-monz

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> Hi Kurt (and Gene)!
>
> Sorry to keep butting in here, just a few points:
>
> > Yes, basically but you gave more options including an explicit
> > mapping, and you used the term "kernel" instead of "comma".
>
> A kernel is actually the set of all the commas that vanish
> in a temperament -- which I think also includes things like
> powers of vanishing commas, and therefore contains an infinite
> number of commas.

yep.

each "vanishing comma" (vapro) is also a promo, which
is simply the ratio which vanishes in a temperament,
along with all of its powers, which also vanish.

all vapros are promos, but not all promos are vapros.
the distinction is that a promo need not vanish.

if a JI tuning is set up so that a certain "comma"
is considered a unison, it most definitely does not
vanish in JI, but if all of its powers are still also
considered unisons (as in standard periodicity-blocks),
then they all together constitute a promo.

in cases where powers of unison-vectors are *not*
considered to be unisons, as for example in torsional-blocks
like the Helmholtz-24 and Groven-36 schismic tunings, then
certain ones of those powers are simply unison-vectors
and not members of a promo, and the other powers
(such as the "comma" itself if one of the higher powers
is a unison-vector) are simply valid scale degrees.

-monz

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> > I'll let Gene answer this, but note that there is also
> > such a thing as a scale (apart from a tempered scale) and
> > it's importance is something I have stressed in our
> > discussions. But I can't find Gene's definition just now,
> > in the Encyclopaedia or on his site.
>
> That definition was much hated. What about this:
>
> A discrete set of real numbers including 0 giving the
> value, in cents, of frequency ratios relative to the base
> frequency represented by 0. A periodic scale has an indexing
> mapping s from the integers and a positive integer n such
> that s[i+n]-s[i] is a constant for all i. A finite scale
> is a finite set of cents values, including 0. Scales can
> also be given multipliciatively, including using rational
> numbers only, so long as the corresponding values in cents
> satisfy the requirements for being a scale.
>
> Note that using this definition, the p-limit for any
> odd prime p is not a scale, but any equal division of
> the octave will be.

that seems like it might be a great definition for a
mathematician ... but for a musician, i think the first
thing that should be mentioned in a definition of "scale"
is that it is a collection of musical pitches.

sure, the set of reals *represents* those pitches,
but the actual physical scale *is* notes, not numbers.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> As far as regular temperaments are concerned, one doesn't get to
> tune the pitches independently. Tuning pitches independently gives
> you what Gene would call a circulating temperament.

Only if it circulates!

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> sure, the set of reals *represents* those pitches,
> but the actual physical scale *is* notes, not numbers.

That means nothing in the Scala scale archive is a scale, since none
of them tell you what the notes are. Almost nothing which has been
posted on the tuning list over the last ten years, and called a scale,
actually is if we follow this definition.

>> As far as regular temperaments are concerned, one doesn't get to
>> tune the pitches independently. Tuning pitches independently gives
>> you what Gene would call a circulating temperament.
>
>Only if it circulates!

So whaddya call an unequal scale which is playable in all keys
but which doesn't have all the good keys lumped together on the
chain of fifths?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> As far as regular temperaments are concerned, one doesn't get to
> >> tune the pitches independently. Tuning pitches independently gives
> >> you what Gene would call a circulating temperament.
> >
> >Only if it circulates!
>
> So whaddya call an unequal scale which is playable in all keys
> but which doesn't have all the good keys lumped together on the
> chain of fifths?

I call it circulating. I think someone (Parizek?) suggested that
should be circular.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> in cases where powers of unison-vectors are *not*
> considered to be unisons, as for example in torsional-blocks
> like the Helmholtz-24 and Groven-36 schismic tunings, then
> certain ones of those powers are simply unison-vectors
> and not members of a promo, and the other powers
> (such as the "comma" itself if one of the higher powers
> is a unison-vector) are simply valid scale degrees.

more correctly i should have said:

... the other powers (such as the "comma" itself if
one of the higher powers is a unison-vector) are simply
valid degrees of the tuning-system.

then i thought to add this:

scales would be smaller subsets of both of these tunings,
but usually would not include a pair of notes separated
by the "comma", but only one of them.

... so at least in terms of deriving smaller subset scales
from these tuning-systems, the "comma" *does* become a sort
of unison-vector after all, which means that the "comma"
and all of its powers *are* in some sense considered
unisons, which means that they do in the end form a promo.

in terms of the temperament as a tuning-system, they
do not form a promo, because these tunings must be
classified as torsional-blocks and not regular
periodicity-blocks, and certain powers of the "comma"
are needed to distinguish the torsional divisions.

-monz

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > sure, the set of reals *represents* those pitches,
> > but the actual physical scale *is* notes, not numbers.
>
> That means nothing in the Scala scale archive is a scale,
> since none of them tell you what the notes are. Almost
> nothing which has been posted on the tuning list over the
> last ten years, and called a scale, actually is if we follow
> this definition.

well, ok, i sure can see that point.

-monz

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

/tuning/topicId_55471.html#55537

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > if i had somebody around who could have given me
> > answers like that 10 years ago, i would have saved
> > myself a ridiculous amount of time and effort.
>
> Yes Monz,
>
> As I've said many times, I don't know what we'd do without your
> encyclopedia. But don't you think that when an encyclopedia that's
> edited by one person starts having entries for new terms added at
> the whim of that editor and deleted again within a few days and new
> terms put in their place, it seriously undermines the authority of
> that encyclopedia. Why should anyone take any notice of any of it
if
> that sort of thing can happen?

***With my apologies to my friend "the Monz," I intend to agree with
Dave here. Monz is intrigued by "exotica"... see his historical
early-civilization websites... Quite frankly, I think "exotica"
should not be a part of terminology and rather than "colorful"
creates an alternate club... a kind of tuning speakeasy (easyspeak??)

J. Pehrson

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > sure, the set of reals *represents* those pitches,
> > but the actual physical scale *is* notes, not numbers.
>
> That means nothing in the Scala scale archive is a scale,

Yes. That is true, of course. Is that somehow suprising?

But of course it would be too tedious to say "representation of a
scale" all the time when everyone knows that's what they are (or at
least I _thought_ everyone knew), so we just call them scales. No
problem.

But when defining what a scale is, surely we want to mention that
it's an actual musical thing with notes (which can be represented in
many different ways), not merely the collection of numbers which
constitute one particular mathematical representation.

One shouldn't mistake the map for the territory, especially not a
single kind of map.

> since none
> of them tell you what the notes are.

But they do tell you what the notes are, once you know how the
representation works.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> But they do tell you what the notes are, once you know how the
> representation works.

No they don't. You need to specify how many Hertz 1/1 is. Should that
really be part of the definition of a scale?

on 8/17/04 4:37 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
>> But they do tell you what the notes are, once you know how the
>> representation works.
>
> No they don't. You need to specify how many Hertz 1/1 is. Should that
> really be part of the definition of a scale?

Well, obviously there's a scale and there's a scale. If I'm not worried
about subtleties of how absolute pitch affects perception of a scale then I
consider the scale to be the same object regardless of where I put 1/1.
That may be a matter of personal preference in how the term is used, and
perhaps I am technically incorrect, yet there seems to be a bit of precedent
for calling something that specifies relative pitches in cents variously
(loosely) a scale, a tuning, a temperament.

In any case the absolute pitch distinction is an orthogonal distinction from
the other distinctions made between temperament, tuning, scale, although I
imagine some would argue that only a scale has an absolute pitch reference.

So do we continue to be loose about scales that are relative versus
absolute, or do we need an adjective here?

-Kurt

on 8/17/04 1:00 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>>> As far as regular temperaments are concerned, one doesn't get to
>>>> tune the pitches independently. Tuning pitches independently gives
>>>> you what Gene would call a circulating temperament.
>>>
>>> Only if it circulates!
>>
>> So whaddya call an unequal scale which is playable in all keys
>> but which doesn't have all the good keys lumped together on the
>> chain of fifths?
>
> I call it circulating. I think someone (Parizek?) suggested that
> should be circular.

No I don't think he intended any distinction between circulating and
circular. He thought I might have intended such a distinction and I
corrected him.

Why would something that isn't at all round be called circular? If anything
"circulating" would be the less specific term if the two would be
distinguished, i.e. the answer to Carl's question should be "circulating"
and the name of the more specific kind of scale in which the good keys are
grouped together and there is a gradual progression should be called
"circular". But I'm not advocating making this distinction without knowing
a bit more about how the terms "circulating" and "circular" are currently
used in the literature in various languages. It would appear that
"circular" and "circulating" might need to be kept synonymous. Maybe some
speakers of non-english languages can confirm whether their word for
"circulating" is something they translate in english as "circular".

One of my own first temperament designs was a 12-tone well-temperered scale
in which the fifth size was the et fifth plus a sin function of the
circle-of-fifths position at some phase offset and some amplitude (I don't
have the details here). Therefore the third size variation was also a sin
function with a slightly different phase. Plotting the two against each
other around the circle of fifths would yield an ellipse. It occurs to me
that plotting the relation 3rd size vs 5th size around the circle of fifths
might in the case of many well-behaved circulating temperaments yield a
closed figure whose shape might reveal something interesting about a scale.
It might also be better to plot on non-orthogonal axes to make the result
tend to be more round rather than flattish.

-Kurt

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > But they do tell you what the notes are, once you know how the
> > representation works.
>
> No they don't. You need to specify how many Hertz 1/1 is.

You're right. But they do tell you what the scale is, which is
really the point.

> Should that
> really be part of the definition of a scale?

In general, no.

But occasionally the author of the scale says it is, and I think
this is sometimes reported in the comment at the start of the .scl
file.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> >
> > > But they do tell you what the notes are, once you know how the
> > > representation works.
> >
> > No they don't. You need to specify how many Hertz 1/1 is.
>
> You're right. But they do tell you what the scale is, which is
> really the point.

Only if you don't define "scale" your way.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> /tuning/topicId_55471.html#55537
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > > if i had somebody around who could have given me
> > > answers like that 10 years ago, i would have saved
> > > myself a ridiculous amount of time and effort.
> >
> > Yes Monz,
> >
> > As I've said many times, I don't know what we'd do without your
> > encyclopedia. But don't you think that when an encyclopedia that's
> > edited by one person starts having entries for new terms added at
> > the whim of that editor and deleted again within a few days and
new
> > terms put in their place, it seriously undermines the authority of
> > that encyclopedia. Why should anyone take any notice of any of it
> if
> > that sort of thing can happen?
>
>
> ***With my apologies to my friend "the Monz," I intend to agree with
> Dave here. Monz is intrigued by "exotica"... see his historical
> early-civilization websites... Quite frankly, I think "exotica"
> should not be a part of terminology and rather than "colorful"
> creates an alternate club... a kind of tuning speakeasy (easyspeak??
)
>
> J. Pehrson

it's an encyclopaedia, so i will attempt to put
*everything* about tuning in it.

anyone who reads it is free to choose what they
read and what they don't read.

as i continue to learn more and more about
tuning-theory and history, it will also be added
into the software in future upgrades. and i'm
going to try as hard as i can to integrate
the Encyclopaedia and the software together.

hey, i'm an artist and the Encyclopaedia and
Musica are together my biggest and most important
work, and they'll both continue to evolve as long
as i'm able to work on them.

someone (probably Dave?) suggested that i mark
the usage of words: "obsolete", "archaic", etc.
that's a good idea ... it would take a while to
do that for all the pages which already exist,
but it's no problem to do it with new ones as i
create them.

i started to get that rolling by redoing the old
"um" and "vum" pages, so now, since those two
terms are archived quite heavily for a couple
of weeks worth of posts in this list, people can
look them up and be referred to their replacements.
that is much better than just having them disappear.

-monz

Hi, Monz,

on 8/18/04 12:18 PM, monz <monz@tonalsoft.com> wrote:

>>> As I've said many times, I don't know what we'd do without your
>>> encyclopedia. But don't you think that when an encyclopedia that's
>>> edited by one person starts having entries for new terms added at
>>> the whim of that editor and deleted again within a few days and
> new
>>> terms put in their place, it seriously undermines the authority of
>>> that encyclopedia. Why should anyone take any notice of any of it
>> if
>>> that sort of thing can happen?
>>
>> ***With my apologies to my friend "the Monz," I intend to agree with
>> Dave here. Monz is intrigued by "exotica"... see his historical
>> early-civilization websites... Quite frankly, I think "exotica"
>> should not be a part of terminology and rather than "colorful"
>> creates an alternate club... a kind of tuning speakeasy (easyspeak??
>
> it's an encyclopaedia, so i will attempt to put
> *everything* about tuning in it.
>
> anyone who reads it is free to choose what they
> read and what they don't read.

People inherently trust a source such as an encyclopedia (any source which
claims "authority") unless they learn to distrust it. So being free to
chose what to read is not very relevant to the "newbie" etc.

And so your following idea sounds good...

> someone (probably Dave?) suggested that i mark
> the usage of words: "obsolete", "archaic", etc.
> that's a good idea

and if you could extend the list to include "tentative", "experimental" etc.
that would take care of the issue.

In fact it might even be good to have an approximate date for each term, or
perhaps each definition in case a term has several definitions from
different centuries, etc.

-Kurt

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Hi, Monz,
>
> on 8/18/04 12:18 PM, monz <monz@t...> wrote:
>
> > someone (probably Dave?) suggested that i mark
> > the usage of words: "obsolete", "archaic", etc.
> > that's a good idea
>
> and if you could extend the list to include
> "tentative", "experimental" etc.
> that would take care of the issue.

yes, i'm going to start doing that kind of thing.

or at least explain with a sentence in the body of
the definition that the term was invented by so-and-so
at such-and-such date.

> In fact it might even be good to have an approximate date
> for each term, or perhaps each definition in case a term
> has several definitions from different centuries, etc.

there's one thing that i have always been very careful
to do: when i copy something from a tuning list or
tuning-math list post, i make sure to include the message
number and date information.

as the years have passed, i've gotten into the habit
of documenting everything in the Encyclopaedia better.
these days, when Paul helps me fix an erroneous definition,
i put "with Paul Erlich" in the update line at the bottom.

-monz